K11a330

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K11a329.gif

K11a329

K11a331.gif

K11a331

Contents

K11a330.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a330 at Knotilus!



Knot presentations

Planar diagram presentation X6271 X12,4,13,3 X18,5,19,6 X22,8,1,7 X16,9,17,10 X4,12,5,11 X20,13,21,14 X8,15,9,16 X10,17,11,18 X2,19,3,20 X14,21,15,22
Gauss code 1, -10, 2, -6, 3, -1, 4, -8, 5, -9, 6, -2, 7, -11, 8, -5, 9, -3, 10, -7, 11, -4
Dowker-Thistlethwaite code 6 12 18 22 16 4 20 8 10 2 14
A Braid Representative
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A Morse Link Presentation K11a330 ML.gif

Three dimensional invariants

Symmetry type Chiral
Unknotting number 2
3-genus 4
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a330/ThurstonBennequinNumber
Hyperbolic Volume 14.4187
A-Polynomial See Data:K11a330/A-polynomial

[edit Notes for K11a330's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a330's four dimensional invariants]

Polynomial invariants

Alexander polynomial t^4-5 t^3+12 t^2-17 t+19-17 t^{-1} +12 t^{-2} -5 t^{-3} + t^{-4}
Conway polynomial z^8+3 z^6+2 z^4+2 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 89, -4 }
Jones polynomial q^2-3 q+6-9 q^{-1} +12 q^{-2} -13 q^{-3} +14 q^{-4} -12 q^{-5} +9 q^{-6} -6 q^{-7} +3 q^{-8} - q^{-9}
HOMFLY-PT polynomial (db, data sources) a^4 z^8-a^6 z^6+6 a^4 z^6-2 a^2 z^6-4 a^6 z^4+14 a^4 z^4-9 a^2 z^4+z^4-5 a^6 z^2+16 a^4 z^2-12 a^2 z^2+3 z^2-3 a^6+7 a^4-5 a^2+2
Kauffman polynomial (db, data sources) z^3 a^{11}+3 z^4 a^{10}+6 z^5 a^9-4 z^3 a^9+z a^9+9 z^6 a^8-12 z^4 a^8+3 z^2 a^8+11 z^7 a^7-22 z^5 a^7+9 z^3 a^7-2 z a^7+11 z^8 a^6-30 z^6 a^6+23 z^4 a^6-12 z^2 a^6+3 a^6+7 z^9 a^5-16 z^7 a^5-3 z^5 a^5+13 z^3 a^5-4 z a^5+2 z^{10} a^4+7 z^8 a^4-52 z^6 a^4+72 z^4 a^4-35 z^2 a^4+7 a^4+10 z^9 a^3-42 z^7 a^3+49 z^5 a^3-14 z^3 a^3+2 z^{10} a^2-3 z^8 a^2-18 z^6 a^2+43 z^4 a^2-27 z^2 a^2+5 a^2+3 z^9 a-15 z^7 a+24 z^5 a-13 z^3 a+z a+z^8-5 z^6+9 z^4-7 z^2+2
The A2 invariant -q^{26}+q^{24}-2 q^{22}-q^{16}+4 q^{14}-q^{12}+3 q^{10}-q^6+q^4-2 q^2+1+ q^{-6}
The G2 invariant Data:K11a330/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a40,}

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

Vassiliev invariants

V2 and V3: (2, -5)
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 -40 32 \frac{364}{3} -\frac{4}{3} -320 -\frac{1744}{3} \frac{128}{3} -168 \frac{256}{3} 800 \frac{2912}{3} -\frac{32}{3} \frac{43351}{15} -\frac{7444}{15} \frac{64564}{45} \frac{905}{9} \frac{2311}{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 K11a330. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-7-6-5-4-3-2-101234χ
5           11
3          2 -2
1         41 3
-1        52  -3
-3       74   3
-5      76    -1
-7     76     1
-9    57      2
-11   47       -3
-13  25        3
-15 14         -3
-17 2          2
-191           -1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-5 i=-3
r=-7 {\mathbb Z}
r=-6 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-5 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-4 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-3 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-2 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=-1 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=0 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{7}
r=1 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=2 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=3 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=4 {\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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K11a329.gif

K11a329

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K11a331