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

Visit K11n167 at Knotilus!

Knot presentations

Planar diagram presentation X6271 X10,4,11,3 X14,6,15,5 X20,8,21,7 X2,10,3,9 X11,18,12,19 X4,14,5,13 X22,15,1,16 X17,12,18,13 X8,20,9,19 X16,21,17,22
Gauss code 1, -5, 2, -7, 3, -1, 4, -10, 5, -2, -6, 9, 7, -3, 8, -11, -9, 6, 10, -4, 11, -8
Dowker-Thistlethwaite code 6 10 14 20 2 -18 4 22 -12 8 16
A Braid Representative
A Morse Link Presentation K11n167 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number \{1,2\}
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11n167/ThurstonBennequinNumber
Hyperbolic Volume 14.2087
A-Polynomial See Data:K11n167/A-polynomial

[edit Notes for K11n167's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 3
Rasmussen s-Invariant -2

[edit Notes for K11n167's four dimensional invariants]

Polynomial invariants

Alexander polynomial t^3-5 t^2+15 t-21+15 t^{-1} -5 t^{-2} + t^{-3}
Conway polynomial z^6+z^4+4 z^2+1
2nd Alexander ideal (db, data sources) \{3,t+1\}
Determinant and Signature { 63, 2 }
Jones polynomial -q^8+3 q^7-7 q^6+9 q^5-10 q^4+12 q^3-9 q^2+7 q-4+ q^{-1}
HOMFLY-PT polynomial (db, data sources) z^6 a^{-4} -2 z^4 a^{-2} +4 z^4 a^{-4} -z^4 a^{-6} -3 z^2 a^{-2} +8 z^2 a^{-4} -2 z^2 a^{-6} +z^2- a^{-2} +5 a^{-4} -3 a^{-6}
Kauffman polynomial (db, data sources) z^9 a^{-3} +z^9 a^{-5} +z^8 a^{-2} +5 z^8 a^{-4} +4 z^8 a^{-6} +z^7 a^{-3} +6 z^7 a^{-5} +5 z^7 a^{-7} +z^6 a^{-2} -8 z^6 a^{-4} -6 z^6 a^{-6} +3 z^6 a^{-8} +4 z^5 a^{-1} -z^5 a^{-3} -16 z^5 a^{-5} -10 z^5 a^{-7} +z^5 a^{-9} +7 z^4 a^{-4} +3 z^4 a^{-6} -5 z^4 a^{-8} +z^4-4 z^3 a^{-1} +4 z^3 a^{-3} +16 z^3 a^{-5} +6 z^3 a^{-7} -2 z^3 a^{-9} -z^2 a^{-2} -5 z^2 a^{-4} -4 z^2 a^{-6} +z^2 a^{-8} -z^2-z a^{-1} -4 z a^{-3} -8 z a^{-5} -4 z a^{-7} +z a^{-9} + a^{-2} +5 a^{-4} +3 a^{-6}
The A2 invariant Data:K11n167/QuantumInvariant/A2/1,0
The G2 invariant Data:K11n167/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11n146,}

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

Vassiliev invariants

V2 and V3: (4, 7)
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 56 128 \frac{872}{3} \frac{160}{3} 896 \frac{4784}{3} \frac{704}{3} 312 \frac{2048}{3} 1568 \frac{13952}{3} \frac{2560}{3} \frac{132422}{15} -\frac{7408}{15} \frac{196688}{45} \frac{922}{9} \frac{9782}{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=2 is the signature of K11n167. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
17         1-1
15        2 2
13       51 -4
11      42  2
9     65   -1
7    64    2
5   36     3
3  46      -2
1 14       3
-1 3        -3
-31         1
Integral Khovanov Homology

(db, data source)

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

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