K11a101

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K11a100.gif

K11a100

K11a102.gif

K11a102

Contents

K11a101.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a101 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X10,4,11,3 X14,6,15,5 X12,7,13,8 X2,10,3,9 X20,11,21,12 X18,14,19,13 X22,16,1,15 X6,18,7,17 X8,19,9,20 X16,22,17,21
Gauss code 1, -5, 2, -1, 3, -9, 4, -10, 5, -2, 6, -4, 7, -3, 8, -11, 9, -7, 10, -6, 11, -8
Dowker-Thistlethwaite code 4 10 14 12 2 20 18 22 6 8 16
A Braid Representative
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A Morse Link Presentation K11a101 ML.gif

Three dimensional invariants

Symmetry type Chiral
Unknotting number 1
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a101/ThurstonBennequinNumber
Hyperbolic Volume 18.521
A-Polynomial See Data:K11a101/A-polynomial

[edit Notes for K11a101's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 3
Rasmussen s-Invariant -2

[edit Notes for K11a101's four dimensional invariants]

Polynomial invariants

Alexander polynomial 3 t^3-16 t^2+39 t-51+39 t^{-1} -16 t^{-2} +3 t^{-3}
Conway polynomial 3 z^6+2 z^4+2 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 167, 2 }
Jones polynomial q^9-5 q^8+11 q^7-18 q^6+24 q^5-27 q^4+27 q^3-23 q^2+17 q-9+4 q^{-1} - q^{-2}
HOMFLY-PT polynomial (db, data sources) z^6 a^{-2} +2 z^6 a^{-4} +z^4 a^{-2} +5 z^4 a^{-4} -3 z^4 a^{-6} -z^4+z^2 a^{-2} +5 z^2 a^{-4} -4 z^2 a^{-6} +z^2 a^{-8} -z^2+ a^{-2} + a^{-4} - a^{-6}
Kauffman polynomial (db, data sources) 3 z^{10} a^{-4} +3 z^{10} a^{-6} +9 z^9 a^{-3} +18 z^9 a^{-5} +9 z^9 a^{-7} +11 z^8 a^{-2} +20 z^8 a^{-4} +19 z^8 a^{-6} +10 z^8 a^{-8} +8 z^7 a^{-1} -4 z^7 a^{-3} -26 z^7 a^{-5} -9 z^7 a^{-7} +5 z^7 a^{-9} -15 z^6 a^{-2} -54 z^6 a^{-4} -57 z^6 a^{-6} -21 z^6 a^{-8} +z^6 a^{-10} +4 z^6+a z^5-10 z^5 a^{-1} -12 z^5 a^{-3} -5 z^5 a^{-5} -13 z^5 a^{-7} -9 z^5 a^{-9} +9 z^4 a^{-2} +44 z^4 a^{-4} +43 z^4 a^{-6} +12 z^4 a^{-8} -z^4 a^{-10} -5 z^4-a z^3+5 z^3 a^{-1} +13 z^3 a^{-3} +16 z^3 a^{-5} +13 z^3 a^{-7} +4 z^3 a^{-9} -z^2 a^{-2} -13 z^2 a^{-4} -12 z^2 a^{-6} -2 z^2 a^{-8} +2 z^2-z a^{-1} -3 z a^{-3} -5 z a^{-5} -3 z a^{-7} - a^{-2} + a^{-4} + a^{-6}
The A2 invariant Data:K11a101/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a101/QuantumInvariant/G2/1,0

"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, 3)
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 24 32 \frac{220}{3} -\frac{4}{3} 192 304 64 -8 \frac{256}{3} 288 \frac{1760}{3} -\frac{32}{3} \frac{18991}{15} \frac{3116}{15} \frac{10564}{45} -\frac{319}{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=2 is the signature of K11a101. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-3-2-1012345678χ
19           11
17          4 -4
15         71 6
13        114  -7
11       137   6
9      1411    -3
7     1313     0
5    1014      4
3   713       -6
1  311        8
-1 16         -5
-3 3          3
-51           -1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=1 i=3
r=-3 {\mathbb Z}
r=-2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-1 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=0 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{7}
r=1 {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=2 {\mathbb Z}^{14}\oplus{\mathbb Z}_2^{13} {\mathbb Z}^{13}
r=3 {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{14} {\mathbb Z}^{14}
r=4 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{13} {\mathbb Z}^{13}
r=5 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=6 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=7 {\mathbb Z}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=8 {\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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K11a100.gif

K11a100

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K11a102