K11a72

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K11a71.gif

K11a71

K11a73.gif

K11a73

Contents

K11a72.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a72 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X10,4,11,3 X12,5,13,6 X14,8,15,7 X2,10,3,9 X22,11,1,12 X18,14,19,13 X20,15,21,16 X8,18,9,17 X6,19,7,20 X16,21,17,22
Gauss code 1, -5, 2, -1, 3, -10, 4, -9, 5, -2, 6, -3, 7, -4, 8, -11, 9, -7, 10, -8, 11, -6
Dowker-Thistlethwaite code 4 10 12 14 2 22 18 20 8 6 16
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gif
BraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart3.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gif
A Morse Link Presentation K11a72 ML.gif

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:K11a72/ThurstonBennequinNumber
Hyperbolic Volume 17.173
A-Polynomial See Data:K11a72/A-polynomial

[edit Notes for K11a72's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus [2,4]
Rasmussen s-Invariant 0

[edit Notes for K11a72's four dimensional invariants]

Polynomial invariants

Alexander polynomial t^4-6 t^3+18 t^2-32 t+39-32 t^{-1} +18 t^{-2} -6 t^{-3} + t^{-4}
Conway polynomial z^8+2 z^6+2 z^4+2 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 153, 0 }
Jones polynomial q^6-5 q^5+10 q^4-16 q^3+22 q^2-24 q+25-21 q^{-1} +15 q^{-2} -9 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^4-6 z^4 a^{-2} +z^4 a^{-4} +10 z^4-3 a^2 z^2-4 z^2 a^{-2} +z^2 a^{-4} +8 z^2-a^2+ a^{-2} - a^{-4} +2
Kauffman polynomial (db, data sources) 2 z^{10} a^{-2} +2 z^{10}+7 a z^9+14 z^9 a^{-1} +7 z^9 a^{-3} +10 a^2 z^8+19 z^8 a^{-2} +9 z^8 a^{-4} +20 z^8+8 a^3 z^7+a z^7-16 z^7 a^{-1} -4 z^7 a^{-3} +5 z^7 a^{-5} +4 a^4 z^6-15 a^2 z^6-55 z^6 a^{-2} -20 z^6 a^{-4} +z^6 a^{-6} -53 z^6+a^5 z^5-12 a^3 z^5-20 a z^5-15 z^5 a^{-1} -18 z^5 a^{-3} -10 z^5 a^{-5} -5 a^4 z^4+11 a^2 z^4+44 z^4 a^{-2} +12 z^4 a^{-4} -z^4 a^{-6} +47 z^4-a^5 z^3+7 a^3 z^3+20 a z^3+22 z^3 a^{-1} +14 z^3 a^{-3} +4 z^3 a^{-5} +a^4 z^2-5 a^2 z^2-12 z^2 a^{-2} -2 z^2 a^{-4} -16 z^2-2 a^3 z-5 a z-5 z a^{-1} -z a^{-3} +z a^{-5} +a^2- a^{-2} - a^{-4} +2
The A2 invariant -q^{14}+2 q^{12}-3 q^{10}+2 q^8+q^6-4 q^4+5 q^2-4+4 q^{-2} + q^{-4} +5 q^{-8} -4 q^{-10} + q^{-12} - q^{-14} -2 q^{-16} + q^{-18}
The G2 invariant q^{80}-3 q^{78}+7 q^{76}-13 q^{74}+16 q^{72}-16 q^{70}+7 q^{68}+16 q^{66}-45 q^{64}+83 q^{62}-114 q^{60}+115 q^{58}-80 q^{56}-9 q^{54}+139 q^{52}-278 q^{50}+389 q^{48}-410 q^{46}+295 q^{44}-41 q^{42}-306 q^{40}+643 q^{38}-839 q^{36}+789 q^{34}-473 q^{32}-53 q^{30}+605 q^{28}-976 q^{26}+1019 q^{24}-679 q^{22}+96 q^{20}+491 q^{18}-837 q^{16}+777 q^{14}-340 q^{12}-279 q^{10}+797 q^8-952 q^6+644 q^4+28 q^2-796+1345 q^{-2} -1425 q^{-4} +976 q^{-6} -156 q^{-8} -748 q^{-10} +1412 q^{-12} -1590 q^{-14} +1240 q^{-16} -491 q^{-18} -358 q^{-20} +997 q^{-22} -1192 q^{-24} +906 q^{-26} -281 q^{-28} -387 q^{-30} +813 q^{-32} -819 q^{-34} +412 q^{-36} +228 q^{-38} -798 q^{-40} +1054 q^{-42} -878 q^{-44} +334 q^{-46} +339 q^{-48} -892 q^{-50} +1112 q^{-52} -949 q^{-54} +500 q^{-56} +44 q^{-58} -492 q^{-60} +703 q^{-62} -662 q^{-64} +443 q^{-66} -155 q^{-68} -92 q^{-70} +226 q^{-72} -253 q^{-74} +196 q^{-76} -106 q^{-78} +32 q^{-80} +21 q^{-82} -39 q^{-84} +35 q^{-86} -24 q^{-88} +11 q^{-90} -4 q^{-92} + q^{-94}

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

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

Vassiliev invariants

V2 and V3: (2, 1)
V2,1 through V6,9:
V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9
8 8 32 \frac{124}{3} -\frac{4}{3} 64 \frac{368}{3} \frac{128}{3} 8 \frac{256}{3} 32 \frac{992}{3} -\frac{32}{3} \frac{6751}{15} -\frac{164}{15} \frac{7684}{45} -\frac{31}{9} \frac{271}{15}

V2,1 through V6,9 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.

Khovanov Homology

The coefficients of the monomials t^rq^j are shown, along with their alternating sums \chi (fixed j, alternation over r). The squares with yellow highlighting are those on the "critical diagonals", where j-2r=s+1 or j-2r=s-1, where s=0 is the signature of K11a72. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-5-4-3-2-10123456χ
13           11
11          4 -4
9         61 5
7        104  -6
5       126   6
3      1210    -2
1     1312     1
-1    913      4
-3   612       -6
-5  39        6
-7 16         -5
-9 3          3
-111           -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}^{6}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-2 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=-1 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=0 {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{13}
r=1 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
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}^{4}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=5 {\mathbb Z}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=6 {\mathbb Z}_2 {\mathbb Z}

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

Modifying This Page

Read me first: Modifying Knot Pages.

See/edit the Hoste-Thistlethwaite Knot Page master template (intermediate).

See/edit the Hoste-Thistlethwaite_Splice_Base (expert).

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