K11a264

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K11a263.gif

K11a263

K11a265.gif

K11a265

Contents

K11a264.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a264 at Knotilus!



Knot presentations

Planar diagram presentation X6271 X8394 X16,6,17,5 X14,7,15,8 X4,9,5,10 X18,12,19,11 X20,14,21,13 X2,16,3,15 X22,17,1,18 X12,20,13,19 X10,22,11,21
Gauss code 1, -8, 2, -5, 3, -1, 4, -2, 5, -11, 6, -10, 7, -4, 8, -3, 9, -6, 10, -7, 11, -9
Dowker-Thistlethwaite code 6 8 16 14 4 18 20 2 22 12 10
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart3.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart3.gif
BraidPart0.gifBraidPart4.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart4.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart0.gif
A Morse Link Presentation K11a264 ML.gif

Three dimensional invariants

Symmetry type Chiral
Unknotting number 1
3-genus 4
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a264/ThurstonBennequinNumber
Hyperbolic Volume 16.2353
A-Polynomial See Data:K11a264/A-polynomial

[edit Notes for K11a264's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus [2,4]
Rasmussen s-Invariant -2

[edit Notes for K11a264's four dimensional invariants]

Polynomial invariants

Alexander polynomial -t^4+6 t^3-16 t^2+28 t-33+28 t^{-1} -16 t^{-2} +6 t^{-3} - t^{-4}
Conway polynomial -z^8-2 z^6+2 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 135, 2 }
Jones polynomial -q^8+4 q^7-9 q^6+14 q^5-19 q^4+22 q^3-21 q^2+19 q-13+8 q^{-1} -4 q^{-2} + q^{-3}
HOMFLY-PT polynomial (db, data sources) -z^8 a^{-2} -5 z^6 a^{-2} +2 z^6 a^{-4} +z^6-9 z^4 a^{-2} +7 z^4 a^{-4} -z^4 a^{-6} +3 z^4-5 z^2 a^{-2} +7 z^2 a^{-4} -2 z^2 a^{-6} +2 z^2+ a^{-2} + a^{-4} - a^{-6}
Kauffman polynomial (db, data sources) 2 z^{10} a^{-2} +2 z^{10} a^{-4} +6 z^9 a^{-1} +12 z^9 a^{-3} +6 z^9 a^{-5} +11 z^8 a^{-2} +13 z^8 a^{-4} +9 z^8 a^{-6} +7 z^8+4 a z^7-10 z^7 a^{-1} -22 z^7 a^{-3} +8 z^7 a^{-7} +a^2 z^6-39 z^6 a^{-2} -36 z^6 a^{-4} -13 z^6 a^{-6} +4 z^6 a^{-8} -19 z^6-10 a z^5+10 z^5 a^{-3} -14 z^5 a^{-5} -13 z^5 a^{-7} +z^5 a^{-9} -2 a^2 z^4+40 z^4 a^{-2} +34 z^4 a^{-4} +7 z^4 a^{-6} -5 z^4 a^{-8} +16 z^4+5 a z^3+4 z^3 a^{-1} +2 z^3 a^{-3} +11 z^3 a^{-5} +7 z^3 a^{-7} -z^3 a^{-9} -13 z^2 a^{-2} -13 z^2 a^{-4} -4 z^2 a^{-6} +z^2 a^{-8} -5 z^2-2 z a^{-5} -2 z a^{-7} - a^{-2} + a^{-4} + a^{-6}
The A2 invariant q^8-2 q^6+2 q^4-2 q^2-1+4 q^{-2} -3 q^{-4} +6 q^{-6} - q^{-8} + q^{-10} + q^{-12} -4 q^{-14} +3 q^{-16} -2 q^{-18} + q^{-22} - q^{-24}
The G2 invariant Data:K11a264/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a157, K11a305,}

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

Vassiliev invariants

V2 and V3: (2, 3)
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
8 24 32 \frac{268}{3} \frac{44}{3} 192 368 0 120 \frac{256}{3} 288 \frac{2144}{3} \frac{352}{3} \frac{22951}{15} -\frac{1468}{5} \frac{38404}{45} \frac{1001}{9} \frac{1831}{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=2 is the signature of K11a264. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-4-3-2-101234567χ
17           1-1
15          3 3
13         61 -5
11        83  5
9       116   -5
7      118    3
5     1011     1
3    911      -2
1   511       6
-1  38        -5
-3 15         4
-5 3          -3
-71           1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=1 i=3
r=-4 {\mathbb Z}
r=-3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-2 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-1 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=0 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{9}
r=1 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=2 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=3 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=4 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=5 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=6 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=7 {\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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K11a263.gif

K11a263

K11a265.gif

K11a265