K11a304

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K11a303.gif

K11a303

K11a305.gif

K11a305

Contents

K11a304.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a304 at Knotilus!



Knot presentations

Planar diagram presentation X6271 X10,4,11,3 X20,6,21,5 X14,8,15,7 X2,10,3,9 X18,11,19,12 X8,14,9,13 X22,16,1,15 X12,17,13,18 X4,20,5,19 X16,22,17,21
Gauss code 1, -5, 2, -10, 3, -1, 4, -7, 5, -2, 6, -9, 7, -4, 8, -11, 9, -6, 10, -3, 11, -8
Dowker-Thistlethwaite code 6 10 20 14 2 18 8 22 12 4 16
A Braid Representative
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A Morse Link Presentation K11a304 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number \{2,3\}
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a304/ThurstonBennequinNumber
Hyperbolic Volume 16.2483
A-Polynomial See Data:K11a304/A-polynomial

[edit Notes for K11a304'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 K11a304's four dimensional invariants]

Polynomial invariants

Alexander polynomial -3 t^3+14 t^2-26 t+31-26 t^{-1} +14 t^{-2} -3 t^{-3}
Conway polynomial -3 z^6-4 z^4+3 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 117, 4 }
Jones polynomial -q^{11}+3 q^{10}-7 q^9+12 q^8-16 q^7+19 q^6-19 q^5+16 q^4-12 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} -7 z^4 a^{-6} +3 z^4 a^{-8} +2 z^2 a^{-2} +3 z^2 a^{-4} -9 z^2 a^{-6} +8 z^2 a^{-8} -z^2 a^{-10} + a^{-2} +2 a^{-4} -5 a^{-6} +5 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} +7 z^8 a^{-6} +9 z^8 a^{-8} +7 z^8 a^{-10} +3 z^7 a^{-3} -9 z^7 a^{-5} -28 z^7 a^{-7} -11 z^7 a^{-9} +5 z^7 a^{-11} +z^6 a^{-2} -11 z^6 a^{-4} -33 z^6 a^{-6} -39 z^6 a^{-8} -15 z^6 a^{-10} +3 z^6 a^{-12} -7 z^5 a^{-3} +2 z^5 a^{-5} +25 z^5 a^{-7} +8 z^5 a^{-9} -7 z^5 a^{-11} +z^5 a^{-13} -3 z^4 a^{-2} +6 z^4 a^{-4} +44 z^4 a^{-6} +57 z^4 a^{-8} +17 z^4 a^{-10} -5 z^4 a^{-12} +3 z^3 a^{-3} -z^3 a^{-5} -4 z^3 a^{-7} +3 z^3 a^{-9} +z^3 a^{-11} -2 z^3 a^{-13} +3 z^2 a^{-2} -6 z^2 a^{-4} -26 z^2 a^{-6} -29 z^2 a^{-8} -11 z^2 a^{-10} +z^2 a^{-12} -2 z a^{-9} -z a^{-11} +z a^{-13} - a^{-2} +2 a^{-4} +5 a^{-6} +5 a^{-8} +2 a^{-10}
The A2 invariant Data:K11a304/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a304/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: (3, 8)
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
12 64 72 366 66 768 \frac{6208}{3} \frac{1120}{3} 320 288 2048 4392 792 \frac{119391}{10} \frac{898}{5} \frac{75502}{15} \frac{577}{6} \frac{6271}{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 K11a304. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-2-10123456789χ
23           1-1
21          2 2
19         51 -4
17        72  5
15       95   -4
13      107    3
11     99     0
9    710      -3
7   59       4
5  37        -4
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}^{7}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=2 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=3 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=4 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=5 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=6 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=7 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
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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K11a303.gif

K11a303

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K11a305