K11a275

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K11a274.gif

K11a274

K11a276.gif

K11a276

Contents

K11a275.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a275 at Knotilus!



Knot presentations

Planar diagram presentation X6271 X10,4,11,3 X14,6,15,5 X20,8,21,7 X2,10,3,9 X18,12,19,11 X4,14,5,13 X22,15,1,16 X12,18,13,17 X8,20,9,19 X16,21,17,22
Gauss code 1, -5, 2, -7, 3, -1, 4, -10, 5, -2, 6, -9, 7, -3, 8, -11, 9, -6, 10, -4, 11, -8
Dowker-Thistlethwaite code 6 10 14 20 2 18 4 22 12 8 16
A Braid Representative
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A Morse Link Presentation K11a275 ML.gif

Three dimensional invariants

Symmetry type Chiral
Unknotting number 2
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a275/ThurstonBennequinNumber
Hyperbolic Volume 16.9486
A-Polynomial See Data:K11a275/A-polynomial

[edit Notes for K11a275'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 K11a275's four dimensional invariants]

Polynomial invariants

Alexander polynomial -3 t^3+15 t^2-29 t+35-29 t^{-1} +15 t^{-2} -3 t^{-3}
Conway polynomial -3 z^6-3 z^4+4 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 129, 4 }
Jones polynomial -q^{11}+3 q^{10}-8 q^9+14 q^8-18 q^7+21 q^6-21 q^5+18 q^4-13 q^3+8 q^2-3 q+1
HOMFLY-PT polynomial (db, data sources) -z^6 a^{-4} -2 z^6 a^{-6} +z^4 a^{-2} -z^4 a^{-4} -6 z^4 a^{-6} +3 z^4 a^{-8} +2 z^2 a^{-2} +2 z^2 a^{-4} -6 z^2 a^{-6} +7 z^2 a^{-8} -z^2 a^{-10} + a^{-2} + a^{-4} -3 a^{-6} +4 a^{-8} -2 a^{-10}
Kauffman polynomial (db, data sources) 2 z^{10} a^{-6} +2 z^{10} a^{-8} +5 z^9 a^{-5} +11 z^9 a^{-7} +6 z^9 a^{-9} +5 z^8 a^{-4} +8 z^8 a^{-6} +11 z^8 a^{-8} +8 z^8 a^{-10} +3 z^7 a^{-3} -7 z^7 a^{-5} -20 z^7 a^{-7} -4 z^7 a^{-9} +6 z^7 a^{-11} +z^6 a^{-2} -10 z^6 a^{-4} -26 z^6 a^{-6} -32 z^6 a^{-8} -14 z^6 a^{-10} +3 z^6 a^{-12} -7 z^5 a^{-3} +z^5 a^{-5} +13 z^5 a^{-7} -5 z^5 a^{-9} -9 z^5 a^{-11} +z^5 a^{-13} -3 z^4 a^{-2} +5 z^4 a^{-4} +25 z^4 a^{-6} +36 z^4 a^{-8} +15 z^4 a^{-10} -4 z^4 a^{-12} +4 z^3 a^{-3} -2 z^3 a^{-5} -6 z^3 a^{-7} +8 z^3 a^{-9} +6 z^3 a^{-11} -2 z^3 a^{-13} +3 z^2 a^{-2} -2 z^2 a^{-4} -14 z^2 a^{-6} -20 z^2 a^{-8} -10 z^2 a^{-10} +z^2 a^{-12} +z a^{-5} +2 z a^{-7} -2 z a^{-9} -2 z a^{-11} +z a^{-13} - a^{-2} + a^{-4} +3 a^{-6} +4 a^{-8} +2 a^{-10}
The A2 invariant Data:K11a275/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a275/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a344,}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {}

Vassiliev invariants

V2 and V3: (4, 11)
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
16 88 128 \frac{1544}{3} \frac{256}{3} 1408 \frac{9712}{3} \frac{1600}{3} 536 \frac{2048}{3} 3872 \frac{24704}{3} \frac{4096}{3} \frac{311702}{15} -\frac{6608}{15} \frac{419408}{45} \frac{2986}{9} \frac{18662}{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=4 is the signature of K11a275. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-2-10123456789χ
23           1-1
21          2 2
19         61 -5
17        82  6
15       106   -4
13      118    3
11     1010     0
9    811      -3
7   510       5
5  38        -5
3 16         5
1 2          -2
-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}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=0 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{3}
r=1 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=2 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=3 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=4 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=5 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=6 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=7 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=8 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
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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K11a274.gif

K11a274

K11a276.gif

K11a276