K11a94

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K11a93.gif

K11a93

K11a95.gif

K11a95

Contents

K11a94.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a94 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X10,4,11,3 X12,6,13,5 X18,8,19,7 X2,10,3,9 X8,12,9,11 X20,14,21,13 X22,16,1,15 X6,18,7,17 X16,20,17,19 X14,22,15,21
Gauss code 1, -5, 2, -1, 3, -9, 4, -6, 5, -2, 6, -3, 7, -11, 8, -10, 9, -4, 10, -7, 11, -8
Dowker-Thistlethwaite code 4 10 12 18 2 8 20 22 6 16 14
A Braid Representative
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A Morse Link Presentation K11a94 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number 3
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a94/ThurstonBennequinNumber
Hyperbolic Volume 14.458
A-Polynomial See Data:K11a94/A-polynomial

[edit Notes for K11a94's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 3
Rasmussen s-Invariant -6

[edit Notes for K11a94's four dimensional invariants]

Polynomial invariants

Alexander polynomial 4 t^3-13 t^2+23 t-27+23 t^{-1} -13 t^{-2} +4 t^{-3}
Conway polynomial 4 z^6+11 z^4+7 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 107, 6 }
Jones polynomial -q^{14}+4 q^{13}-8 q^{12}+12 q^{11}-16 q^{10}+17 q^9-17 q^8+14 q^7-9 q^6+6 q^5-2 q^4+q^3
HOMFLY-PT polynomial (db, data sources) z^6 a^{-6} +2 z^6 a^{-8} +z^6 a^{-10} +4 z^4 a^{-6} +7 z^4 a^{-8} +z^4 a^{-10} -z^4 a^{-12} +5 z^2 a^{-6} +7 z^2 a^{-8} -4 z^2 a^{-10} -z^2 a^{-12} +2 a^{-6} +2 a^{-8} -4 a^{-10} + a^{-12}
Kauffman polynomial (db, data sources) z^{10} a^{-10} +z^{10} a^{-12} +2 z^9 a^{-9} +6 z^9 a^{-11} +4 z^9 a^{-13} +3 z^8 a^{-8} +5 z^8 a^{-10} +9 z^8 a^{-12} +7 z^8 a^{-14} +2 z^7 a^{-7} +2 z^7 a^{-9} -6 z^7 a^{-11} +z^7 a^{-13} +7 z^7 a^{-15} +z^6 a^{-6} -7 z^6 a^{-8} -13 z^6 a^{-10} -19 z^6 a^{-12} -10 z^6 a^{-14} +4 z^6 a^{-16} -5 z^5 a^{-7} -14 z^5 a^{-9} -7 z^5 a^{-11} -11 z^5 a^{-13} -12 z^5 a^{-15} +z^5 a^{-17} -4 z^4 a^{-6} +6 z^4 a^{-8} +13 z^4 a^{-10} +12 z^4 a^{-12} +3 z^4 a^{-14} -6 z^4 a^{-16} +2 z^3 a^{-7} +15 z^3 a^{-9} +17 z^3 a^{-11} +10 z^3 a^{-13} +5 z^3 a^{-15} -z^3 a^{-17} +5 z^2 a^{-6} -5 z^2 a^{-8} -10 z^2 a^{-10} -z^2 a^{-12} +z^2 a^{-16} +z a^{-7} -6 z a^{-9} -9 z a^{-11} -3 z a^{-13} -z a^{-15} -2 a^{-6} +2 a^{-8} +4 a^{-10} + a^{-12}
The A2 invariant Data:K11a94/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a94/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: (7, 17)
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
28 136 392 \frac{2546}{3} \frac{310}{3} 3808 \frac{17872}{3} \frac{2848}{3} 616 \frac{10976}{3} 9248 \frac{71288}{3} \frac{8680}{3} \frac{1301737}{30} \frac{13682}{5} \frac{600554}{45} \frac{5399}{18} \frac{46537}{30}

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=6 is the signature of K11a94. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
01234567891011χ
29           1-1
27          3 3
25         51 -4
23        73  4
21       95   -4
19      87    1
17     99     0
15    58      -3
13   49       5
11  25        -3
9  4         4
712          -1
51           1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=5 i=7
r=0 {\mathbb Z} {\mathbb Z}
r=1 {\mathbb Z}^{2}
r=2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=3 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=4 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=5 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=6 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=7 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=8 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=9 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=10 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
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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K11a93

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K11a95