K11a353

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K11a352.gif

K11a352

K11a354.gif

K11a354

Contents

K11a353.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a353 at Knotilus!


Knot K11a353.
A graph, K11a353.
A part of a link and a part of a graph.

Knot presentations

Planar diagram presentation X8291 X12,4,13,3 X16,6,17,5 X18,8,19,7 X22,10,1,9 X4,12,5,11 X20,14,21,13 X2,16,3,15 X6,18,7,17 X10,20,11,19 X14,22,15,21
Gauss code 1, -8, 2, -6, 3, -9, 4, -1, 5, -10, 6, -2, 7, -11, 8, -3, 9, -4, 10, -7, 11, -5
Dowker-Thistlethwaite code 8 12 16 18 22 4 20 2 6 10 14
A Braid Representative
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A Morse Link Presentation K11a353 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number \{3,4\}
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a353/ThurstonBennequinNumber
Hyperbolic Volume 16.151
A-Polynomial See Data:K11a353/A-polynomial

[edit Notes for K11a353's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 3
Rasmussen s-Invariant -6

[edit Notes for K11a353's four dimensional invariants]

Polynomial invariants

Alexander polynomial 5 t^3-15 t^2+26 t-31+26 t^{-1} -15 t^{-2} +5 t^{-3}
Conway polynomial 5 z^6+15 z^4+11 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 123, 6 }
Jones polynomial -q^{14}+3 q^{13}-8 q^{12}+13 q^{11}-17 q^{10}+20 q^9-20 q^8+17 q^7-12 q^6+8 q^5-3 q^4+q^3
HOMFLY-PT polynomial (db, data sources) z^6 a^{-6} +3 z^6 a^{-8} +z^6 a^{-10} +3 z^4 a^{-6} +12 z^4 a^{-8} +z^4 a^{-10} -z^4 a^{-12} +2 z^2 a^{-6} +14 z^2 a^{-8} -3 z^2 a^{-10} -2 z^2 a^{-12} +4 a^{-8} -2 a^{-10} - a^{-12}
Kauffman polynomial (db, data sources) 2 z^{10} a^{-10} +2 z^{10} a^{-12} +5 z^9 a^{-9} +10 z^9 a^{-11} +5 z^9 a^{-13} +6 z^8 a^{-8} +6 z^8 a^{-10} +7 z^8 a^{-12} +7 z^8 a^{-14} +3 z^7 a^{-7} -8 z^7 a^{-9} -18 z^7 a^{-11} -z^7 a^{-13} +6 z^7 a^{-15} +z^6 a^{-6} -18 z^6 a^{-8} -21 z^6 a^{-10} -15 z^6 a^{-12} -10 z^6 a^{-14} +3 z^6 a^{-16} -7 z^5 a^{-7} -z^5 a^{-9} +8 z^5 a^{-11} -9 z^5 a^{-13} -10 z^5 a^{-15} +z^5 a^{-17} -3 z^4 a^{-6} +22 z^4 a^{-8} +19 z^4 a^{-10} +5 z^4 a^{-12} +7 z^4 a^{-14} -4 z^4 a^{-16} +3 z^3 a^{-7} +6 z^3 a^{-9} +3 z^3 a^{-11} +10 z^3 a^{-13} +8 z^3 a^{-15} -2 z^3 a^{-17} +2 z^2 a^{-6} -16 z^2 a^{-8} -9 z^2 a^{-10} +4 z^2 a^{-12} -4 z^2 a^{-14} +z^2 a^{-16} -3 z a^{-9} -z a^{-11} -3 z a^{-13} -4 z a^{-15} +z a^{-17} +4 a^{-8} +2 a^{-10} - a^{-12}
The A2 invariant Data:K11a353/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a353/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: (11, 35)
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
44 280 968 \frac{7018}{3} \frac{1070}{3} 12320 \frac{64816}{3} \frac{11392}{3} 2840 \frac{42592}{3} 39200 \frac{308792}{3} \frac{47080}{3} \frac{6099941}{30} \frac{28666}{5} \frac{3532402}{45} \frac{25051}{18} \frac{306821}{30}

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=6 is the signature of K11a353. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
01234567891011χ
29           1-1
27          2 2
25         61 -5
23        72  5
21       106   -4
19      107    3
17     1010     0
15    710      -3
13   510       5
11  37        -4
9  5         5
713          -2
51           1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=5 i=7
r=0 {\mathbb Z} {\mathbb Z}
r=1 {\mathbb Z}^{3}
r=2 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=3 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=4 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=5 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=6 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=7 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=8 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=9 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
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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K11a352.gif

K11a352

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K11a354