K11a320

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K11a319.gif

K11a319

K11a321.gif

K11a321

Contents

K11a320.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a320 at Knotilus!



Knot presentations

Planar diagram presentation X6271 X12,4,13,3 X16,6,17,5 X22,8,1,7 X18,10,19,9 X4,12,5,11 X20,14,21,13 X2,16,3,15 X10,18,11,17 X8,20,9,19 X14,22,15,21
Gauss code 1, -8, 2, -6, 3, -1, 4, -10, 5, -9, 6, -2, 7, -11, 8, -3, 9, -5, 10, -7, 11, -4
Dowker-Thistlethwaite code 6 12 16 22 18 4 20 2 10 8 14
A Braid Representative
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A Morse Link Presentation K11a320 ML.gif

Three dimensional invariants

Symmetry type Chiral
Unknotting number \{2,3\}
3-genus 2
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a320/ThurstonBennequinNumber
Hyperbolic Volume 15.7364
A-Polynomial See Data:K11a320/A-polynomial

[edit Notes for K11a320'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 K11a320's four dimensional invariants]

Polynomial invariants

Alexander polynomial 9 t^2-27 t+37-27 t^{-1} +9 t^{-2}
Conway polynomial 9 z^4+9 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 109, 4 }
Jones polynomial -q^{13}+3 q^{12}-7 q^{11}+11 q^{10}-15 q^9+17 q^8-17 q^7+16 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} +7 z^2 a^{-8} -2 z^2 a^{-10} -z^2 a^{-12} + a^{-6} +3 a^{-8} -3 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} +8 z^8 a^{-10} +2 z^8 a^{-12} +3 z^8 a^{-14} +8 z^7 a^{-7} -6 z^7 a^{-9} -27 z^7 a^{-11} -12 z^7 a^{-13} +z^7 a^{-15} +6 z^6 a^{-6} -17 z^6 a^{-8} -33 z^6 a^{-10} -21 z^6 a^{-12} -11 z^6 a^{-14} +3 z^5 a^{-5} -10 z^5 a^{-7} -8 z^5 a^{-9} +17 z^5 a^{-11} +8 z^5 a^{-13} -4 z^5 a^{-15} +z^4 a^{-4} -7 z^4 a^{-6} +16 z^4 a^{-8} +32 z^4 a^{-10} +20 z^4 a^{-12} +12 z^4 a^{-14} -2 z^3 a^{-5} +5 z^3 a^{-7} +11 z^3 a^{-9} -2 z^3 a^{-11} -z^3 a^{-13} +5 z^3 a^{-15} -z^2 a^{-4} +5 z^2 a^{-6} -9 z^2 a^{-8} -15 z^2 a^{-10} -4 z^2 a^{-12} -4 z^2 a^{-14} -4 z a^{-9} -z a^{-11} +z a^{-13} -2 z a^{-15} - a^{-6} +3 a^{-8} +3 a^{-10}
The A2 invariant Data:K11a320/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a320/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: (9, 26)
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
36 208 648 1578 246 7488 \frac{39616}{3} \frac{6880}{3} 1808 7776 21632 56808 8856 \frac{1125053}{10} \frac{11254}{5} \frac{222782}{5} \frac{1761}{2} \frac{58973}{10}

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 K11a320. 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        62  4
19       95   -4
17      86    2
15     99     0
13    78      -1
11   49       5
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}^{9}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=5 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=6 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=7 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=8 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
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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K11a319.gif

K11a319

K11a321.gif

K11a321