K11a173

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K11a172.gif

K11a172

K11a174.gif

K11a174

Contents

K11a173.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a173 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X10,4,11,3 X20,6,21,5 X14,8,15,7 X2,10,3,9 X8,12,9,11 X18,13,19,14 X22,15,1,16 X6,18,7,17 X12,19,13,20 X16,21,17,22
Gauss code 1, -5, 2, -1, 3, -9, 4, -6, 5, -2, 6, -10, 7, -4, 8, -11, 9, -7, 10, -3, 11, -8
Dowker-Thistlethwaite code 4 10 20 14 2 8 18 22 6 12 16
A Braid Representative
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A Morse Link Presentation K11a173 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:K11a173/ThurstonBennequinNumber
Hyperbolic Volume 16.7456
A-Polynomial See Data:K11a173/A-polynomial

[edit Notes for K11a173'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 K11a173's four dimensional invariants]

Polynomial invariants

Alexander polynomial 2 t^3-12 t^2+32 t-43+32 t^{-1} -12 t^{-2} +2 t^{-3}
Conway polynomial 2 z^6+2 z^2+1
2nd Alexander ideal (db, data sources) \{3,t+1\}
Determinant and Signature { 135, 2 }
Jones polynomial -q^8+4 q^7-10 q^6+15 q^5-19 q^4+23 q^3-21 q^2+18 q-13+7 q^{-1} -3 q^{-2} + q^{-3}
HOMFLY-PT polynomial (db, data sources) z^6 a^{-2} +z^6 a^{-4} +z^4 a^{-2} +2 z^4 a^{-4} -z^4 a^{-6} -2 z^4+a^2 z^2+z^2 a^{-2} +4 z^2 a^{-4} -z^2 a^{-6} -3 z^2+a^2+ a^{-2} +3 a^{-4} -2 a^{-6} -2
Kauffman polynomial (db, data sources) 2 z^{10} a^{-2} +2 z^{10} a^{-4} +4 z^9 a^{-1} +11 z^9 a^{-3} +7 z^9 a^{-5} +6 z^8 a^{-2} +13 z^8 a^{-4} +11 z^8 a^{-6} +4 z^8+3 a z^7-2 z^7 a^{-1} -17 z^7 a^{-3} -3 z^7 a^{-5} +9 z^7 a^{-7} +a^2 z^6-13 z^6 a^{-2} -31 z^6 a^{-4} -20 z^6 a^{-6} +4 z^6 a^{-8} -5 z^6-8 a z^5-8 z^5 a^{-1} +5 z^5 a^{-3} -11 z^5 a^{-5} -15 z^5 a^{-7} +z^5 a^{-9} -3 a^2 z^4+21 z^4 a^{-4} +15 z^4 a^{-6} -4 z^4 a^{-8} -5 z^4+7 a z^3+8 z^3 a^{-1} +4 z^3 a^{-3} +13 z^3 a^{-5} +9 z^3 a^{-7} -z^3 a^{-9} +3 a^2 z^2+6 z^2 a^{-2} -5 z^2 a^{-4} -6 z^2 a^{-6} +8 z^2-2 a z-4 z a^{-1} -4 z a^{-3} -6 z a^{-5} -4 z a^{-7} -a^2- a^{-2} +3 a^{-4} +2 a^{-6} -2
The A2 invariant Data:K11a173/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a173/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: (2, 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
8 40 32 \frac{364}{3} \frac{44}{3} 320 \frac{1936}{3} \frac{64}{3} 200 \frac{256}{3} 800 \frac{2912}{3} \frac{352}{3} \frac{45271}{15} -\frac{7124}{15} \frac{77284}{45} \frac{857}{9} \frac{3511}{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 K11a173. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-4-3-2-101234567χ
17           1-1
15          3 3
13         71 -6
11        83  5
9       117   -4
7      128    4
5     911     2
3    912      -3
1   510       5
-1  28        -6
-3 15         4
-5 2          -2
-71           1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=1 i=3
r=-4 {\mathbb Z}
r=-3 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-2 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-1 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=0 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{9}
r=1 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=2 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r=3 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=4 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=5 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=6 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
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

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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K11a172.gif

K11a172

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K11a174