K11a263

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K11a262.gif

K11a262

K11a264.gif

K11a264

Contents

K11a263.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a263 at Knotilus!



Knot presentations

Planar diagram presentation X6271 X8493 X16,6,17,5 X2837 X20,10,21,9 X22,12,1,11 X18,14,19,13 X4,16,5,15 X12,18,13,17 X14,20,15,19 X10,22,11,21
Gauss code 1, -4, 2, -8, 3, -1, 4, -2, 5, -11, 6, -9, 7, -10, 8, -3, 9, -7, 10, -5, 11, -6
Dowker-Thistlethwaite code 6 8 16 2 20 22 18 4 12 14 10
A Braid Representative
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A Morse Link Presentation K11a263 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number 4
3-genus 4
Bridge index 4
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a263/ThurstonBennequinNumber
Hyperbolic Volume 13.7204
A-Polynomial See Data:K11a263/A-polynomial

[edit Notes for K11a263's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 4
Rasmussen s-Invariant -8

[edit Notes for K11a263's four dimensional invariants]

Polynomial invariants

Alexander polynomial 2 t^4-6 t^3+11 t^2-14 t+15-14 t^{-1} +11 t^{-2} -6 t^{-3} +2 t^{-4}
Conway polynomial 2 z^8+10 z^6+15 z^4+8 z^2+1
2nd Alexander ideal (db, data sources) \left\{t^2-t+1\right\}
Determinant and Signature { 81, 8 }
Jones polynomial -q^{15}+3 q^{14}-6 q^{13}+10 q^{12}-12 q^{11}+12 q^{10}-13 q^9+10 q^8-7 q^7+5 q^6-q^5+q^4
HOMFLY-PT polynomial (db, data sources) z^8 a^{-8} +z^8 a^{-10} +7 z^6 a^{-8} +4 z^6 a^{-10} -z^6 a^{-12} +18 z^4 a^{-8} -3 z^4 a^{-12} +20 z^2 a^{-8} -13 z^2 a^{-10} +z^2 a^{-12} +8 a^{-8} -11 a^{-10} +5 a^{-12} - a^{-14}
Kauffman polynomial (db, data sources) z^{10} a^{-10} +z^{10} a^{-12} +z^9 a^{-9} +5 z^9 a^{-11} +4 z^9 a^{-13} +z^8 a^{-8} -z^8 a^{-10} +7 z^8 a^{-12} +9 z^8 a^{-14} -3 z^7 a^{-9} -14 z^7 a^{-11} +z^7 a^{-13} +12 z^7 a^{-15} -7 z^6 a^{-8} -11 z^6 a^{-10} -32 z^6 a^{-12} -18 z^6 a^{-14} +10 z^6 a^{-16} -3 z^5 a^{-9} -6 z^5 a^{-11} -36 z^5 a^{-13} -27 z^5 a^{-15} +6 z^5 a^{-17} +18 z^4 a^{-8} +28 z^4 a^{-10} +28 z^4 a^{-12} -15 z^4 a^{-16} +3 z^4 a^{-18} +16 z^3 a^{-9} +37 z^3 a^{-11} +43 z^3 a^{-13} +18 z^3 a^{-15} -3 z^3 a^{-17} +z^3 a^{-19} -20 z^2 a^{-8} -26 z^2 a^{-10} -7 z^2 a^{-12} +5 z^2 a^{-14} +6 z^2 a^{-16} -12 z a^{-9} -22 z a^{-11} -16 z a^{-13} -6 z a^{-15} +8 a^{-8} +11 a^{-10} +5 a^{-12} + a^{-14}
The A2 invariant  q^{-14} +4 q^{-18} + q^{-20} +4 q^{-22} + q^{-24} -2 q^{-26} - q^{-28} -7 q^{-30} -2 q^{-34} +2 q^{-36} +3 q^{-38} + q^{-42} -2 q^{-44}
The G2 invariant Data:K11a263/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: (8, 21)
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
32 168 512 \frac{3424}{3} \frac{392}{3} 5376 8720 1344 904 \frac{16384}{3} 14112 \frac{109568}{3} \frac{12544}{3} \frac{1035844}{15} \frac{56744}{15} \frac{969736}{45} \frac{5804}{9} \frac{37444}{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=8 is the signature of K11a263. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
01234567891011χ
31           1-1
29          2 2
27         41 -3
25        62  4
23       64   -2
21      66    0
19     76     -1
17    36      -3
15   47       3
13  13        -2
11  4         4
911          0
71           1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=7 i=9
r=0 {\mathbb Z} {\mathbb Z}
r=1 {\mathbb Z}
r=2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2 {\mathbb Z}
r=3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=4 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=5 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=6 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=7 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=8 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=9 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
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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K11a262.gif

K11a262

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K11a264