K11a172

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K11a171.gif

K11a171

K11a173.gif

K11a173

Contents

K11a172.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a172 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X10,3,11,4 X18,5,19,6 X22,8,1,7 X14,10,15,9 X2,11,3,12 X20,14,21,13 X8,16,9,15 X6,17,7,18 X12,20,13,19 X16,22,17,21
Gauss code 1, -6, 2, -1, 3, -9, 4, -8, 5, -2, 6, -10, 7, -5, 8, -11, 9, -3, 10, -7, 11, -4
Dowker-Thistlethwaite code 4 10 18 22 14 2 20 8 6 12 16
A Braid Representative
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A Morse Link Presentation K11a172 ML.gif

Three dimensional invariants

Symmetry type Chiral
Unknotting number \{1,2\}
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a172/ThurstonBennequinNumber
Hyperbolic Volume 17.202
A-Polynomial See Data:K11a172/A-polynomial

[edit Notes for K11a172'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 K11a172's four dimensional invariants]

Polynomial invariants

Alexander polynomial 2 t^3-13 t^2+33 t-43+33 t^{-1} -13 t^{-2} +2 t^{-3}
Conway polynomial 2 z^6-z^4-z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 139, 2 }
Jones polynomial -q^8+4 q^7-9 q^6+15 q^5-20 q^4+23 q^3-22 q^2+19 q-14+8 q^{-1} -3 q^{-2} + q^{-3}
HOMFLY-PT polynomial (db, data sources) z^6 a^{-2} +z^6 a^{-4} +2 z^4 a^{-4} -z^4 a^{-6} -2 z^4+a^2 z^2-3 z^2 a^{-2} +4 z^2 a^{-4} -z^2 a^{-6} -2 z^2+a^2-2 a^{-2} +3 a^{-4} - a^{-6}
Kauffman polynomial (db, data sources) 2 z^{10} a^{-2} +2 z^{10} a^{-4} +5 z^9 a^{-1} +12 z^9 a^{-3} +7 z^9 a^{-5} +10 z^8 a^{-2} +15 z^8 a^{-4} +10 z^8 a^{-6} +5 z^8+3 a z^7-5 z^7 a^{-1} -19 z^7 a^{-3} -3 z^7 a^{-5} +8 z^7 a^{-7} +a^2 z^6-29 z^6 a^{-2} -39 z^6 a^{-4} -16 z^6 a^{-6} +4 z^6 a^{-8} -9 z^6-7 a z^5-3 z^5 a^{-1} +8 z^5 a^{-3} -9 z^5 a^{-5} -12 z^5 a^{-7} +z^5 a^{-9} -3 a^2 z^4+25 z^4 a^{-2} +35 z^4 a^{-4} +11 z^4 a^{-6} -5 z^4 a^{-8} +3 z^4+5 a z^3+z^3 a^{-1} -3 z^3 a^{-3} +8 z^3 a^{-5} +6 z^3 a^{-7} -z^3 a^{-9} +3 a^2 z^2-12 z^2 a^{-2} -15 z^2 a^{-4} -4 z^2 a^{-6} +z^2 a^{-8} +z^2-a z+z a^{-3} -z a^{-5} -z a^{-7} -a^2+2 a^{-2} +3 a^{-4} + a^{-6}
The A2 invariant Data:K11a172/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a172/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a54,}

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

Vassiliev invariants

V2 and V3: (-1, 1)
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
-4 8 8 \frac{130}{3} \frac{38}{3} -32 -\frac{16}{3} -\frac{160}{3} 40 -\frac{32}{3} 32 -\frac{520}{3} -\frac{152}{3} -\frac{4111}{30} -\frac{1406}{5} \frac{6778}{45} \frac{463}{18} \frac{1169}{30}

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 K11a172. 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         61 -5
11        93  6
9       116   -5
7      129    3
5     1011     1
3    912      -3
1   611       5
-1  28        -6
-3 16         5
-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}^{6}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-1 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=0 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{9}
r=1 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=2 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r=3 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=4 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=5 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
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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K11a171

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K11a173