K11a308

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K11a307.gif

K11a307

K11a309.gif

K11a309

Contents

K11a308.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a308 at Knotilus!



Knot presentations

Planar diagram presentation X6271 X12,4,13,3 X14,6,15,5 X18,8,19,7 X20,10,21,9 X2,12,3,11 X4,14,5,13 X22,15,1,16 X10,18,11,17 X8,20,9,19 X16,21,17,22
Gauss code 1, -6, 2, -7, 3, -1, 4, -10, 5, -9, 6, -2, 7, -3, 8, -11, 9, -4, 10, -5, 11, -8
Dowker-Thistlethwaite code 6 12 14 18 20 2 4 22 10 8 16
A Braid Representative
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A Morse Link Presentation K11a308 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number 3
3-genus 4
Bridge index 2
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a308/ThurstonBennequinNumber
Hyperbolic Volume 10.8655
A-Polynomial See Data:K11a308/A-polynomial

[edit Notes for K11a308's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a308's four dimensional invariants]

Polynomial invariants

Alexander polynomial -t^4+5 t^3-9 t^2+13 t-15+13 t^{-1} -9 t^{-2} +5 t^{-3} - t^{-4}
Conway polynomial -z^8-3 z^6+z^4+6 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 71, 6 }
Jones polynomial -q^{12}+3 q^{11}-6 q^{10}+8 q^9-10 q^8+11 q^7-10 q^6+9 q^5-6 q^4+4 q^3-2 q^2+q
HOMFLY-PT polynomial (db, data sources) -z^8 a^{-6} +z^6 a^{-4} -6 z^6 a^{-6} +2 z^6 a^{-8} +5 z^4 a^{-4} -12 z^4 a^{-6} +9 z^4 a^{-8} -z^4 a^{-10} +7 z^2 a^{-4} -9 z^2 a^{-6} +11 z^2 a^{-8} -3 z^2 a^{-10} +2 a^{-4} -2 a^{-6} +3 a^{-8} -2 a^{-10}
Kauffman polynomial (db, data sources) z^{10} a^{-6} +z^{10} a^{-8} +2 z^9 a^{-5} +5 z^9 a^{-7} +3 z^9 a^{-9} +z^8 a^{-4} -z^8 a^{-6} +4 z^8 a^{-8} +6 z^8 a^{-10} -11 z^7 a^{-5} -20 z^7 a^{-7} -z^7 a^{-9} +8 z^7 a^{-11} -6 z^6 a^{-4} -12 z^6 a^{-6} -25 z^6 a^{-8} -11 z^6 a^{-10} +8 z^6 a^{-12} +19 z^5 a^{-5} +21 z^5 a^{-7} -18 z^5 a^{-9} -14 z^5 a^{-11} +6 z^5 a^{-13} +12 z^4 a^{-4} +26 z^4 a^{-6} +30 z^4 a^{-8} +2 z^4 a^{-10} -11 z^4 a^{-12} +3 z^4 a^{-14} -10 z^3 a^{-5} -4 z^3 a^{-7} +17 z^3 a^{-9} +5 z^3 a^{-11} -5 z^3 a^{-13} +z^3 a^{-15} -9 z^2 a^{-4} -14 z^2 a^{-6} -13 z^2 a^{-8} -4 z^2 a^{-10} +4 z^2 a^{-12} -4 z a^{-9} -2 z a^{-11} +2 z a^{-13} +2 a^{-4} +2 a^{-6} +3 a^{-8} +2 a^{-10}
The A2 invariant  q^{-4} + q^{-8} - q^{-12} +2 q^{-14} - q^{-16} +3 q^{-18} + q^{-24} -2 q^{-26} + q^{-28} - q^{-30} - q^{-36}
The G2 invariant Data:K11a308/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: (6, 16)
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
24 128 288 828 132 3072 \frac{17792}{3} \frac{2912}{3} 928 2304 8192 19872 3168 \frac{215311}{5} -\frac{2204}{5} \frac{278204}{15} \frac{1681}{3} \frac{12431}{5}

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 K11a308. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-2-10123456789χ
25           1-1
23          2 2
21         41 -3
19        42  2
17       64   -2
15      54    1
13     56     1
11    45      -1
9   25       3
7  24        -2
5 13         2
3 1          -1
11           1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=5 i=7
r=-2 {\mathbb Z}
r=-1 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=0 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}^{2}
r=1 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=2 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=3 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=4 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=5 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=6 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=7 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
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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K11a307.gif

K11a307

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K11a309