K11a310

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K11a309.gif

K11a309

K11a311.gif

K11a311

Contents

K11a310.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a310 at Knotilus!



Knot presentations

Planar diagram presentation X6271 X12,4,13,3 X14,6,15,5 X20,8,21,7 X18,10,19,9 X2,12,3,11 X4,14,5,13 X22,15,1,16 X10,18,11,17 X8,20,9,19 X16,21,17,22
Gauss code 1, -6, 2, -7, 3, -1, 4, -10, 5, -9, 6, -2, 7, -3, 8, -11, 9, -5, 10, -4, 11, -8
Dowker-Thistlethwaite code 6 12 14 20 18 2 4 22 10 8 16
A Braid Representative
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A Morse Link Presentation K11a310 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number 2
3-genus 3
Bridge index 2
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a310/ThurstonBennequinNumber
Hyperbolic Volume 10.1293
A-Polynomial See Data:K11a310/A-polynomial

[edit Notes for K11a310's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a310's four dimensional invariants]

Polynomial invariants

Alexander polynomial -2 t^3+9 t^2-13 t+13-13 t^{-1} +9 t^{-2} -2 t^{-3}
Conway polynomial -2 z^6-3 z^4+5 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 61, 4 }
Jones polynomial -q^{11}+2 q^{10}-4 q^9+6 q^8-8 q^7+10 q^6-9 q^5+8 q^4-6 q^3+4 q^2-2 q+1
HOMFLY-PT polynomial (db, data sources) -z^6 a^{-4} -z^6 a^{-6} +z^4 a^{-2} -3 z^4 a^{-4} -3 z^4 a^{-6} +2 z^4 a^{-8} +3 z^2 a^{-2} -z^2 a^{-4} -2 z^2 a^{-6} +6 z^2 a^{-8} -z^2 a^{-10} + a^{-2} - a^{-6} +3 a^{-8} -2 a^{-10}
Kauffman polynomial (db, data sources) z^{10} a^{-6} +z^{10} a^{-8} +2 z^9 a^{-5} +4 z^9 a^{-7} +2 z^9 a^{-9} +2 z^8 a^{-4} -2 z^8 a^{-6} -z^8 a^{-8} +3 z^8 a^{-10} +2 z^7 a^{-3} -7 z^7 a^{-5} -17 z^7 a^{-7} -5 z^7 a^{-9} +3 z^7 a^{-11} +z^6 a^{-2} -5 z^6 a^{-4} +z^6 a^{-6} -5 z^6 a^{-8} -10 z^6 a^{-10} +2 z^6 a^{-12} -7 z^5 a^{-3} +11 z^5 a^{-5} +31 z^5 a^{-7} +3 z^5 a^{-9} -9 z^5 a^{-11} +z^5 a^{-13} -4 z^4 a^{-2} +2 z^4 a^{-6} +17 z^4 a^{-8} +14 z^4 a^{-10} -5 z^4 a^{-12} +5 z^3 a^{-3} -11 z^3 a^{-5} -21 z^3 a^{-7} +6 z^3 a^{-9} +8 z^3 a^{-11} -3 z^3 a^{-13} +4 z^2 a^{-2} +2 z^2 a^{-4} -4 z^2 a^{-6} -11 z^2 a^{-8} -8 z^2 a^{-10} +z^2 a^{-12} +2 z a^{-5} +4 z a^{-7} -2 z a^{-9} -3 z a^{-11} +z a^{-13} - a^{-2} + a^{-6} +3 a^{-8} +2 a^{-10}
The A2 invariant Data:K11a310/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a310/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: (5, 14)
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
20 112 200 \frac{2086}{3} \frac{362}{3} 2240 \frac{14368}{3} \frac{2272}{3} 880 \frac{4000}{3} 6272 \frac{41720}{3} \frac{7240}{3} \frac{199135}{6} -\frac{4958}{3} \frac{144590}{9} \frac{10693}{18} \frac{13567}{6}

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 K11a310. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-2-10123456789χ
23           1-1
21          1 1
19         31 -2
17        31  2
15       53   -2
13      53    2
11     45     1
9    45      -1
7   24       2
5  24        -2
3 13         2
1 1          -1
-11           1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=3 i=5
r=-2 {\mathbb Z}
r=-1 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=0 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}^{2}
r=1 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=3 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=4 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=5 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=6 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=7 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=8 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=9 {\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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K11a309.gif

K11a309

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K11a311