K11n175

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K11n174.gif

K11n174

K11n176.gif

K11n176

Contents

K11n175.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11n175 at Knotilus!



Knot presentations

Planar diagram presentation X6271 X3,11,4,10 X5,17,6,16 X14,8,15,7 X9,21,10,20 X11,18,12,19 X13,3,14,2 X22,16,1,15 X17,12,18,13 X19,5,20,4 X21,9,22,8
Gauss code 1, 7, -2, 10, -3, -1, 4, 11, -5, 2, -6, 9, -7, -4, 8, 3, -9, 6, -10, 5, -11, -8
Dowker-Thistlethwaite code 6 -10 -16 14 -20 -18 -2 22 -12 -4 -8
A Braid Representative
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A Morse Link Presentation K11n175 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number 2
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11n175/ThurstonBennequinNumber
Hyperbolic Volume 13.9821
A-Polynomial See Data:K11n175/A-polynomial

[edit Notes for K11n175'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 K11n175's four dimensional invariants]

Polynomial invariants

Alexander polynomial -2 t^3+9 t^2-14 t+15-14 t^{-1} +9 t^{-2} -2 t^{-3}
Conway polynomial -2 z^6-3 z^4+4 z^2+1
2nd Alexander ideal (db, data sources) \left\{2,t^2+t+1\right\}
Determinant and Signature { 65, 4 }
Jones polynomial -2 q^9+5 q^8-8 q^7+10 q^6-11 q^5+11 q^4-8 q^3+6 q^2-3 q+1
HOMFLY-PT polynomial (db, data sources) -z^6 a^{-4} -z^6 a^{-6} +z^4 a^{-2} -2 z^4 a^{-4} -3 z^4 a^{-6} +z^4 a^{-8} +2 z^2 a^{-2} +2 z^2 a^{-4} -3 z^2 a^{-6} +3 z^2 a^{-8} +3 a^{-4} -3 a^{-6} +2 a^{-8} - a^{-10}
Kauffman polynomial (db, data sources) 2 z^9 a^{-5} +2 z^9 a^{-7} +4 z^8 a^{-4} +8 z^8 a^{-6} +4 z^8 a^{-8} +3 z^7 a^{-3} -z^7 a^{-5} -z^7 a^{-7} +3 z^7 a^{-9} +z^6 a^{-2} -12 z^6 a^{-4} -23 z^6 a^{-6} -9 z^6 a^{-8} +z^6 a^{-10} -9 z^5 a^{-3} -6 z^5 a^{-5} +z^5 a^{-7} -2 z^5 a^{-9} -3 z^4 a^{-2} +11 z^4 a^{-4} +25 z^4 a^{-6} +15 z^4 a^{-8} +4 z^4 a^{-10} +6 z^3 a^{-3} +2 z^3 a^{-5} -5 z^3 a^{-7} +2 z^3 a^{-9} +3 z^3 a^{-11} +2 z^2 a^{-2} -7 z^2 a^{-4} -15 z^2 a^{-6} -10 z^2 a^{-8} -4 z^2 a^{-10} +2 z a^{-7} -2 z a^{-11} +3 a^{-4} +3 a^{-6} +2 a^{-8} + a^{-10}
The A2 invariant Data:K11n175/QuantumInvariant/A2/1,0
The G2 invariant Data:K11n175/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {K11n103,}

Vassiliev invariants

V2 and V3: (4, 9)
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
16 72 128 \frac{1256}{3} \frac{256}{3} 1152 2544 448 488 \frac{2048}{3} 2592 \frac{20096}{3} \frac{4096}{3} \frac{230822}{15} -\frac{13568}{15} \frac{358928}{45} \frac{2122}{9} \frac{17702}{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 K11n175. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-2-101234567χ
19         2-2
17        3 3
15       52 -3
13      53  2
11     65   -1
9    55    0
7   36     3
5  35      -2
3 14       3
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}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{3}
r=1 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=2 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=3 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=4 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=5 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=6 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=7 {\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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K11n174.gif

K11n174

K11n176.gif

K11n176