K11a354

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K11a353.gif

K11a353

K11a355.gif

K11a355

Contents

K11a354.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a354 at Knotilus!


Knot K11a354.
A graph, K11a354.

Knot presentations

Planar diagram presentation X8291 X12,4,13,3 X16,6,17,5 X20,8,21,7 X2,10,3,9 X18,12,19,11 X22,14,1,13 X6,16,7,15 X4,18,5,17 X10,20,11,19 X14,22,15,21
Gauss code 1, -5, 2, -9, 3, -8, 4, -1, 5, -10, 6, -2, 7, -11, 8, -3, 9, -6, 10, -4, 11, -7
Dowker-Thistlethwaite code 8 12 16 20 2 18 22 6 4 10 14
A Braid Representative
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A Morse Link Presentation K11a354 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number \{2,3,4\}
3-genus 2
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a354/ThurstonBennequinNumber
Hyperbolic Volume 15.1817
A-Polynomial See Data:K11a354/A-polynomial

[edit Notes for K11a354's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a354's four dimensional invariants]

Polynomial invariants

Alexander polynomial 9 t^2-26 t+35-26 t^{-1} +9 t^{-2}
Conway polynomial 9 z^4+10 z^2+1
2nd Alexander ideal (db, data sources) \left\{2,t^2+t+1\right\}
Determinant and Signature { 105, 4 }
Jones polynomial -q^{13}+3 q^{12}-7 q^{11}+10 q^{10}-14 q^9+17 q^8-16 q^7+15 q^6-11 q^5+7 q^4-3 q^3+q^2
HOMFLY-PT polynomial (db, data sources) z^4 a^{-4} +3 z^4 a^{-6} +4 z^4 a^{-8} +z^4 a^{-10} +z^2 a^{-4} +4 z^2 a^{-6} +8 z^2 a^{-8} -2 z^2 a^{-10} -z^2 a^{-12} +5 a^{-8} -4 a^{-10}
Kauffman polynomial (db, data sources) 2 z^{10} a^{-10} +2 z^{10} a^{-12} +6 z^9 a^{-9} +10 z^9 a^{-11} +4 z^9 a^{-13} +9 z^8 a^{-8} +7 z^8 a^{-10} +z^8 a^{-12} +3 z^8 a^{-14} +8 z^7 a^{-7} -9 z^7 a^{-9} -31 z^7 a^{-11} -13 z^7 a^{-13} +z^7 a^{-15} +6 z^6 a^{-6} -20 z^6 a^{-8} -34 z^6 a^{-10} -19 z^6 a^{-12} -11 z^6 a^{-14} +3 z^5 a^{-5} -11 z^5 a^{-7} +z^5 a^{-9} +30 z^5 a^{-11} +11 z^5 a^{-13} -4 z^5 a^{-15} +z^4 a^{-4} -7 z^4 a^{-6} +25 z^4 a^{-8} +42 z^4 a^{-10} +20 z^4 a^{-12} +11 z^4 a^{-14} -2 z^3 a^{-5} +6 z^3 a^{-7} +3 z^3 a^{-9} -15 z^3 a^{-11} -5 z^3 a^{-13} +5 z^3 a^{-15} -z^2 a^{-4} +4 z^2 a^{-6} -18 z^2 a^{-8} -25 z^2 a^{-10} -5 z^2 a^{-12} -3 z^2 a^{-14} -2 z a^{-9} +4 z a^{-11} +4 z a^{-13} -2 z a^{-15} +5 a^{-8} +4 a^{-10}
The A2 invariant Data:K11a354/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a354/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: (10, 30)
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
40 240 800 \frac{5804}{3} \frac{964}{3} 9600 17056 3008 2480 \frac{32000}{3} 28800 \frac{232160}{3} \frac{38560}{3} \frac{457711}{3} \frac{3860}{3} \frac{573772}{9} \frac{10165}{9} \frac{26095}{3}

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=4 is the signature of K11a354. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
01234567891011χ
27           1-1
25          2 2
23         51 -4
21        52  3
19       95   -4
17      85    3
15     89     1
13    78      -1
11   48       4
9  37        -4
7  4         4
513          -2
31           1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=3 i=5
r=0 {\mathbb Z} {\mathbb Z}
r=1 {\mathbb Z}^{3}
r=2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=3 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=4 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=5 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=6 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=7 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=8 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=9 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=10 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=11 {\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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K11a353.gif

K11a353

K11a355.gif

K11a355