K11a24

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K11a23

K11a25

1 Knot presentations
2 Three dimensional invariants
3 Four dimensional invariants
4 Polynomial invariants
5 "Similar" Knots (within the Atlas)
6 Vassiliev invariants
7 Khovanov Homology
8 Computer Talk
9 Modifying This Page

(Knotscape image)

See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a24's page at Knotilus!

Visit K11a24's page at the original Knot Atlas!

[edit K11a24 Quick Notes]

[edit K11a24 Further Notes and Views]

[edit] Knot presentations

Planar diagram presentation	X₄₂₅₁ X₈₄₉₃ X_12,5,13,6 X₂₈₃₇ X_18,10,19,9 X_16,11,17,12 X_6,13,7,14 X_20,15,21,16 X_22,18,1,17 X_14,19,15,20 X_10,22,11,21
Gauss code	1, -4, 2, -1, 3, -7, 4, -2, 5, -11, 6, -3, 7, -10, 8, -6, 9, -5, 10, -8, 11, -9
Dowker-Thistlethwaite code	4 8 12 2 18 16 6 20 22 14 10

A Braid Representative

A Morse Link Presentation

[edit] Three dimensional invariants

Symmetry type	Chiral
Unknotting number	${1,2}$
3-genus	4
Bridge index	3
Super bridge index	Missing
Nakanishi index	Missing
Maximal Thurston-Bennequin number	Data:K11a24/ThurstonBennequinNumber
Hyperbolic Volume	17.9006
A-Polynomial	See Data:K11a24/A-polynomial

[edit Notes for K11a24's three dimensional invariants]

[edit] Four dimensional invariants

Smooth 4 genus	Missing
Topological 4 genus	Missing
Concordance genus	$4$
Rasmussen s-Invariant	0

[edit Notes for K11a24's four dimensional invariants]

[edit] Polynomial invariants

Alexander polynomial	$t 4 -6 t 3 + 18 t 2 -33 t + 41-33 t -1 + 18 t -2 -6 t -3 + t -4$
Conway polynomial	$z 8 + 2 z 6 + 2 z 4 + z 2 + 1$
2nd Alexander ideal (db, data sources)	${1}$
Determinant and Signature	{ 157, 0 }
Jones polynomial	$q 6 -4 q 5 + 9 q 4 -16 q 3 + 22 q 2 -25 q + 26-22 q -1 + 17 q -2 -10 q -3 + 4 q -4 - q -5$
HOMFLY-PT polynomial (db, data sources)	$z 8 - a 2 z 6 -2 z 6 a -2 + 5 z 6 -3 a 2 z 4 -7 z 4 a -2 + z 4 a -4 + 11 z 4 -4 a 2 z 2 -9 z 2 a -2 + 2 z 2 a -4 + 12 z 2 -2 a 2 -4 a -2 + a -4 + 6$
Kauffman polynomial (db, data sources)	$2 z 10 a -2 + 2 z 10 + 8 a z 9 + 14 z 9 a -1 + 6 z 9 a -3 + 12 a 2 z 8 + 18 z 8 a -2 + 7 z 8 a -4 + 23 z 8 + 9 a 3 z 7 + 2 a z 7 -12 z 7 a -1 - z 7 a -3 + 4 z 7 a -5 + 4 a 4 z 6 -19 a 2 z 6 -50 z 6 a -2 -14 z 6 a -4 + z 6 a -6 -58 z 6 + a 5 z 5 -13 a 3 z 5 -26 a z 5 -24 z 5 a -1 -21 z 5 a -3 -9 z 5 a -5 -4 a 4 z 4 + 15 a 2 z 4 + 42 z 4 a -2 + 9 z 4 a -4 -2 z 4 a -6 + 50 z 4 - a 5 z 3 + 9 a 3 z 3 + 25 a z 3 + 30 z 3 a -1 + 22 z 3 a -3 + 7 z 3 a -5 + a 4 z 2 -8 a 2 z 2 -18 z 2 a -2 -3 z 2 a -4 + z 2 a -6 -23 z 2 -3 a 3 z -8 a z -10 z a -1 -7 z a -3 -2 z a -5 + 2 a 2 + 4 a -2 + a -4 + 6$
The A2 invariant	$- q 14 + 2 q 12 -4 q 10 + 2 q 8 + q 6 -3 q 4 + 7 q 2 -3 + 5 q -2 - q -4 -2 q -6 + 3 q -8 -5 q -10 + 2 q -12 - q -16 + q -18$
The G2 invariant	$q 80 -3 q 78 + 7 q 76 -13 q 74 + 17 q 72 -19 q 70 + 12 q 68 + 11 q 66 -46 q 64 + 94 q 62 -139 q 60 + 154 q 58 -123 q 56 + 20 q 54 + 160 q 52 -369 q 50 + 543 q 48 -584 q 46 + 408 q 44 -30 q 42 -482 q 40 + 949 q 38 -1167 q 36 + 1001 q 34 -454 q 32 -314 q 30 + 1007 q 28 -1337 q 26 + 1155 q 24 -520 q 22 -297 q 20 + 930 q 18 -1100 q 16 + 723 q 14 + 43 q 12 -830 q 10 + 1288 q 8 -1170 q 6 + 494 q 4 + 492 q 2 -1389 + 1846 q -2 -1655 q -4 + 877 q -6 + 226 q -8 -1251 q -10 + 1840 q -12 -1780 q -14 + 1119 q -16 -121 q -18 -823 q -20 + 1338 q -22 -1252 q -24 + 639 q -26 + 208 q -28 -899 q -30 + 1117 q -32 -782 q -34 + 40 q -36 + 763 q -38 -1275 q -40 + 1280 q -42 -790 q -44 + 9 q -46 + 733 q -48 -1173 q -50 + 1190 q -52 -823 q -54 + 270 q -56 + 253 q -58 -591 q -60 + 671 q -62 -539 q -64 + 300 q -66 -52 q -68 -120 q -70 + 194 q -72 -188 q -74 + 134 q -76 -66 q -78 + 16 q -80 + 15 q -82 -26 q -84 + 22 q -86 -16 q -88 + 8 q -90 -3 q -92 + q -94$

Further Quantum Invariants

K11a24 Quantum Invariants.

Computer Talk

The above data is available with the Mathematica package KnotTheory`, as shown in the (simulated) Mathematica session below. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting. This Mathematica session is also available (albeit only for the knot 5_2) as the notebook PolynomialInvariantsSession.nb.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:= AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of August 31, 2006, 11:25:27.5625.

In[3]:= K = Knot["K11a24"];

In[4]:= Alexander[K][t]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]= $t 4 -6 t 3 + 18 t 2 -33 t + 41-33 t -1 + 18 t -2 -6 t -3 + t -4$

In[5]:= Conway[K][z]

Out[5]= $z 8 + 2 z 6 + 2 z 4 + z 2 + 1$

In[6]:= Alexander[K, 2][t]

KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.

Out[6]= ${1}$

In[7]:= {KnotDet[K], KnotSignature[K]}

Out[7]= { 157, 0 }

In[8]:= Jones[K][q]

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[8]= $q 6 -4 q 5 + 9 q 4 -16 q 3 + 22 q 2 -25 q + 26-22 q -1 + 17 q -2 -10 q -3 + 4 q -4 - q -5$

In[9]:= HOMFLYPT[K][a, z]

KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.

Out[9]= $z 8 - a 2 z 6 -2 z 6 a -2 + 5 z 6 -3 a 2 z 4 -7 z 4 a -2 + z 4 a -4 + 11 z 4 -4 a 2 z 2 -9 z 2 a -2 + 2 z 2 a -4 + 12 z 2 -2 a 2 -4 a -2 + a -4 + 6$

In[10]:= Kauffman[K][a, z]

KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.

Out[10]= $2 z 10 a -2 + 2 z 10 + 8 a z 9 + 14 z 9 a -1 + 6 z 9 a -3 + 12 a 2 z 8 + 18 z 8 a -2 + 7 z 8 a -4 + 23 z 8 + 9 a 3 z 7 + 2 a z 7 -12 z 7 a -1 - z 7 a -3 + 4 z 7 a -5 + 4 a 4 z 6 -19 a 2 z 6 -50 z 6 a -2 -14 z 6 a -4 + z 6 a -6 -58 z 6 + a 5 z 5 -13 a 3 z 5 -26 a z 5 -24 z 5 a -1 -21 z 5 a -3 -9 z 5 a -5 -4 a 4 z 4 + 15 a 2 z 4 + 42 z 4 a -2 + 9 z 4 a -4 -2 z 4 a -6 + 50 z 4 - a 5 z 3 + 9 a 3 z 3 + 25 a z 3 + 30 z 3 a -1 + 22 z 3 a -3 + 7 z 3 a -5 + a 4 z 2 -8 a 2 z 2 -18 z 2 a -2 -3 z 2 a -4 + z 2 a -6 -23 z 2 -3 a 3 z -8 a z -10 z a -1 -7 z a -3 -2 z a -5 + 2 a 2 + 4 a -2 + a -4 + 6$

[edit] "Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a26, K11a315,}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$ ): {K11a26, K11a315,}

Computer Talk

The above data is available with the Mathematica package KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:= AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of May 31, 2006, 14:15:20.091.

In[3]:= K = Knot["K11a24"];

In[4]:= {A = Alexander[K][t], J = Jones[K][q]}

KnotTheory::loading: Loading precomputed data in PD4Knots`.

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[4]= { $t 4 -6 t 3 + 18 t 2 -33 t + 41-33 t -1 + 18 t -2 -6 t -3 + t -4$ , $q 6 -4 q 5 + 9 q 4 -16 q 3 + 22 q 2 -25 q + 26-22 q -1 + 17 q -2 -10 q -3 + 4 q -4 - q -5$ }

In[5]:= DeleteCases[Select[AllKnots[], (A === Alexander[#][t]) &], K]

KnotTheory::loading: Loading precomputed data in DTCode4KnotsTo11`.

KnotTheory::credits: The GaussCode to PD conversion was written by Siddarth Sankaran at the University of Toronto in the summer of 2005.

Out[5]= {K11a26, K11a315,}

In[6]:= DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]

KnotTheory::loading: Loading precomputed data in Jones4Knots11`.

Out[6]= {K11a26, K11a315,}

[edit] Vassiliev invariants

V₂ and V₃:

(1, -1)

V_2,1 through V_6,9:

V_2,1

V_3,1

V_4,1

V_4,2

V_4,3

V_5,1

V_5,2

V_5,3

V_5,4

V_6,1

V_6,2

V_6,3

V_6,4

V_6,5

V_6,6

V_6,7

V_6,8

V_6,9

4

-8

8

$-\frac{34}{3}$

$-\frac{38}{3}$

-32

$-\frac{80}{3}$

$\frac{64}{3}$

-8

$\frac{32}{3}$

32

$-\frac{136}{3}$

$-\frac{152}{3}$

$\frac{751}{30}$

$-\frac{902}{15}$

$\frac{3302}{45}$

$-\frac{175}{18}$

$\frac{271}{30}$

V_2,1 through V_6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V₂ and V₃.

[edit] Khovanov Homology

The coefficients of the monomials

t r q j

are shown, along with their alternating sums

χ

(fixed

j

, alternation over

r

). The squares with yellow highlighting are those on the "critical diagonals", where

j -2 r = s + 1

j -2 r = s -1

, where

s =

0 is the signature of K11a24. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\		r
	\
j		\

-5

-4

-3

-2

-1

-3

-7

-3

-1

-3

-5

-7

-6

-9

-11

-1

Integral Khovanov Homology

(db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -1$	$i = 1$
$r = -5$	${\mathbb Z}$
$r = -4$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -3$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = -2$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{7}$	${\mathbb Z}^{7}$
$r = -1$	${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{10}$	${\mathbb Z}^{10}$
$r = 0$	${\mathbb Z}^{14}\oplus{\mathbb Z}_2^{12}$	${\mathbb Z}^{13}$
$r = 1$	${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{13}$	${\mathbb Z}^{13}$
$r = 2$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{12}$	${\mathbb Z}^{12}$
$r = 3$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{10}$	${\mathbb Z}^{10}$
$r = 4$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = 5$	${\mathbb Z}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 6$	${\mathbb Z}_2$	${\mathbb Z}$