K11n136

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K11n135.gif

K11n135

K11n137.gif

K11n137

Contents

K11n136.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11n136 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X12,4,13,3 X5,17,6,16 X14,8,15,7 X18,10,19,9 X2,12,3,11 X8,14,9,13 X15,1,16,22 X20,18,21,17 X10,20,11,19 X21,7,22,6
Gauss code 1, -6, 2, -1, -3, 11, 4, -7, 5, -10, 6, -2, 7, -4, -8, 3, 9, -5, 10, -9, -11, 8
Dowker-Thistlethwaite code 4 12 -16 14 18 2 8 -22 20 10 -6
A Braid Representative
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A Morse Link Presentation K11n136 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:K11n136/ThurstonBennequinNumber
Hyperbolic Volume 13.7948
A-Polynomial See Data:K11n136/A-polynomial

[edit Notes for K11n136'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 K11n136's four dimensional invariants]

Polynomial invariants

Alexander polynomial 3 t^3-8 t^2+13 t-15+13 t^{-1} -8 t^{-2} +3 t^{-3}
Conway polynomial 3 z^6+10 z^4+8 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 63, 6 }
Jones polynomial -2 q^{12}+5 q^{11}-8 q^{10}+10 q^9-11 q^8+10 q^7-8 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} +4 z^4 a^{-6} +8 z^4 a^{-8} -2 z^4 a^{-10} +5 z^2 a^{-6} +8 z^2 a^{-8} -5 z^2 a^{-10} +2 a^{-6} + a^{-8} -2 a^{-10}
Kauffman polynomial (db, data sources) z^9 a^{-9} +z^9 a^{-11} +3 z^8 a^{-8} +6 z^8 a^{-10} +3 z^8 a^{-12} +2 z^7 a^{-7} +5 z^7 a^{-9} +6 z^7 a^{-11} +3 z^7 a^{-13} +z^6 a^{-6} -8 z^6 a^{-8} -12 z^6 a^{-10} -2 z^6 a^{-12} +z^6 a^{-14} -5 z^5 a^{-7} -19 z^5 a^{-9} -16 z^5 a^{-11} -2 z^5 a^{-13} -4 z^4 a^{-6} +7 z^4 a^{-8} +7 z^4 a^{-10} +4 z^4 a^{-14} +2 z^3 a^{-7} +14 z^3 a^{-9} +12 z^3 a^{-11} +3 z^3 a^{-13} +3 z^3 a^{-15} +5 z^2 a^{-6} -5 z^2 a^{-8} -6 z^2 a^{-10} +z^2 a^{-12} -3 z^2 a^{-14} +z a^{-7} -3 z a^{-9} -3 z a^{-11} -z a^{-13} -2 z a^{-15} -2 a^{-6} + a^{-8} +2 a^{-10}
The A2 invariant Data:K11n136/QuantumInvariant/A2/1,0
The G2 invariant Data:K11n136/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, 22)
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
32 176 512 \frac{3712}{3} \frac{512}{3} 5632 \frac{29120}{3} \frac{5024}{3} 1136 \frac{16384}{3} 15488 \frac{118784}{3} \frac{16384}{3} \frac{1171204}{15} \frac{56704}{15} \frac{1231216}{45} \frac{4460}{9} \frac{50644}{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=6 is the signature of K11n136. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
0123456789χ
25         2-2
23        3 3
21       52 -3
19      53  2
17     65   -1
15    45    -1
13   46     2
11  24      -2
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}^{4}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=4 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=5 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=6 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=7 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=8 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=9 {\mathbb Z}_2^{2} {\mathbb Z}^{2}

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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K11n135.gif

K11n135

K11n137.gif

K11n137