K11a178

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K11a177.gif

K11a177

K11a179.gif

K11a179

Contents

K11a178.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a178 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X12,4,13,3 X14,5,15,6 X16,8,17,7 X18,10,19,9 X2,12,3,11 X22,13,1,14 X20,16,21,15 X10,18,11,17 X8,20,9,19 X6,21,7,22
Gauss code 1, -6, 2, -1, 3, -11, 4, -10, 5, -9, 6, -2, 7, -3, 8, -4, 9, -5, 10, -8, 11, -7
Dowker-Thistlethwaite code 4 12 14 16 18 2 22 20 10 8 6
A Braid Representative
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A Morse Link Presentation K11a178 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number 2
3-genus 3
Bridge index 2
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a178/ThurstonBennequinNumber
Hyperbolic Volume 14.6424
A-Polynomial See Data:K11a178/A-polynomial

[edit Notes for K11a178'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 K11a178's four dimensional invariants]

Polynomial invariants

Alexander polynomial 2 t^3-11 t^2+29 t-39+29 t^{-1} -11 t^{-2} +2 t^{-3}
Conway polynomial 2 z^6+z^4+3 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 123, 2 }
Jones polynomial q^9-4 q^8+7 q^7-12 q^6+17 q^5-19 q^4+20 q^3-17 q^2+13 q-8+4 q^{-1} - q^{-2}
HOMFLY-PT polynomial (db, data sources) z^6 a^{-2} +z^6 a^{-4} +2 z^4 a^{-2} +2 z^4 a^{-4} -2 z^4 a^{-6} -z^4+2 z^2 a^{-2} +4 z^2 a^{-4} -3 z^2 a^{-6} +z^2 a^{-8} -z^2+3 a^{-4} -2 a^{-6}
Kauffman polynomial (db, data sources) z^{10} a^{-4} +z^{10} a^{-6} +4 z^9 a^{-3} +8 z^9 a^{-5} +4 z^9 a^{-7} +7 z^8 a^{-2} +14 z^8 a^{-4} +13 z^8 a^{-6} +6 z^8 a^{-8} +7 z^7 a^{-1} +6 z^7 a^{-3} -6 z^7 a^{-5} -z^7 a^{-7} +4 z^7 a^{-9} -7 z^6 a^{-2} -33 z^6 a^{-4} -39 z^6 a^{-6} -16 z^6 a^{-8} +z^6 a^{-10} +4 z^6+a z^5-11 z^5 a^{-1} -22 z^5 a^{-3} -16 z^5 a^{-5} -17 z^5 a^{-7} -11 z^5 a^{-9} -z^4 a^{-2} +29 z^4 a^{-4} +39 z^4 a^{-6} +13 z^4 a^{-8} -2 z^4 a^{-10} -6 z^4-a z^3+5 z^3 a^{-1} +17 z^3 a^{-3} +23 z^3 a^{-5} +19 z^3 a^{-7} +7 z^3 a^{-9} +z^2 a^{-2} -13 z^2 a^{-4} -16 z^2 a^{-6} -4 z^2 a^{-8} +2 z^2-z a^{-1} -4 z a^{-3} -8 z a^{-5} -5 z a^{-7} +3 a^{-4} +2 a^{-6}
The A2 invariant Data:K11a178/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a178/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a294,}

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

Vassiliev invariants

V2 and V3: (3, 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
12 40 72 174 26 480 \frac{2608}{3} \frac{544}{3} 104 288 800 2088 312 \frac{42751}{10} \frac{1774}{15} \frac{8754}{5} \frac{11}{2} \frac{2431}{10}

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 K11a178. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-3-2-1012345678χ
19           11
17          3 -3
15         41 3
13        83  -5
11       94   5
9      108    -2
7     109     1
5    710      3
3   610       -4
1  38        5
-1 15         -4
-3 3          3
-51           -1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=1 i=3
r=-3 {\mathbb Z}
r=-2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-1 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=0 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{6}
r=1 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=2 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=3 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=4 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=5 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=6 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=7 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=8 {\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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K11a177.gif

K11a177

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K11a179