K11n173

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K11n172.gif

K11n172

K11n174.gif

K11n174

Contents

K11n173.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11n173 at Knotilus!



Knot presentations

Planar diagram presentation X6271 X3,11,4,10 X16,6,17,5 X12,8,13,7 X20,10,21,9 X11,3,12,2 X18,13,19,14 X4,16,5,15 X22,17,1,18 X8,20,9,19 X14,21,15,22
Gauss code 1, 6, -2, -8, 3, -1, 4, -10, 5, 2, -6, -4, 7, -11, 8, -3, 9, -7, 10, -5, 11, -9
Dowker-Thistlethwaite code 6 -10 16 12 20 -2 18 4 22 8 14
A Braid Representative
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A Morse Link Presentation K11n173 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number \{2,3\}
3-genus 4
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11n173/ThurstonBennequinNumber
Hyperbolic Volume 13.5804
A-Polynomial See Data:K11n173/A-polynomial

[edit Notes for K11n173's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11n173's four dimensional invariants]

Polynomial invariants

Alexander polynomial t^4-4 t^3+8 t^2-7 t+5-7 t^{-1} +8 t^{-2} -4 t^{-3} + t^{-4}
Conway polynomial z^8+4 z^6+4 z^4+5 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 45, 4 }
Jones polynomial -3 q^7+5 q^6-6 q^5+8 q^4-7 q^3+7 q^2-5 q+3- q^{-1}
HOMFLY-PT polynomial (db, data sources) z^8 a^{-4} -z^6 a^{-2} +6 z^6 a^{-4} -z^6 a^{-6} -4 z^4 a^{-2} +13 z^4 a^{-4} -5 z^4 a^{-6} -4 z^2 a^{-2} +15 z^2 a^{-4} -7 z^2 a^{-6} +z^2 a^{-8} -2 a^{-2} +7 a^{-4} -4 a^{-6}
Kauffman polynomial (db, data sources) 2 z^9 a^{-3} +2 z^9 a^{-5} +3 z^8 a^{-2} +8 z^8 a^{-4} +5 z^8 a^{-6} +z^7 a^{-1} -4 z^7 a^{-3} -2 z^7 a^{-5} +3 z^7 a^{-7} -13 z^6 a^{-2} -34 z^6 a^{-4} -21 z^6 a^{-6} -4 z^5 a^{-1} -9 z^5 a^{-3} -14 z^5 a^{-5} -9 z^5 a^{-7} +16 z^4 a^{-2} +43 z^4 a^{-4} +30 z^4 a^{-6} +3 z^4 a^{-8} +5 z^3 a^{-1} +17 z^3 a^{-3} +20 z^3 a^{-5} +8 z^3 a^{-7} -7 z^2 a^{-2} -25 z^2 a^{-4} -20 z^2 a^{-6} -2 z^2 a^{-8} -2 z a^{-1} -6 z a^{-3} -8 z a^{-5} -z a^{-7} +3 z a^{-9} +2 a^{-2} +7 a^{-4} +4 a^{-6}
The A2 invariant -q^2+1- q^{-2} + q^{-8} +4 q^{-10} +4 q^{-14} - q^{-16} - q^{-20} -2 q^{-22} -2 q^{-26} + q^{-28}
The G2 invariant Data:K11n173/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: (5, 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
20 72 200 \frac{1222}{3} \frac{194}{3} 1440 2416 416 360 \frac{4000}{3} 2592 \frac{24440}{3} \frac{3880}{3} \frac{87967}{6} \frac{170}{3} \frac{55766}{9} \frac{1381}{18} \frac{5119}{6}

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 K11n173. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-3-2-1012345χ
15        3-3
13       2 2
11      43 -1
9     42  2
7    34   1
5   44    0
3  24     2
1 13      -2
-1 2       2
-31        -1
Integral Khovanov Homology

(db, data source)

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

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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K11n172.gif

K11n172

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K11n174