K11a174

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K11a173.gif

K11a173

K11a175.gif

K11a175

Contents

K11a174.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a174 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X12,4,13,3 X14,5,15,6 X16,7,17,8 X18,10,19,9 X2,12,3,11 X22,13,1,14 X6,15,7,16 X20,17,21,18 X8,20,9,19 X10,21,11,22
Gauss code 1, -6, 2, -1, 3, -8, 4, -10, 5, -11, 6, -2, 7, -3, 8, -4, 9, -5, 10, -9, 11, -7
Dowker-Thistlethwaite code 4 12 14 16 18 2 22 6 20 8 10
A Braid Representative
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A Morse Link Presentation K11a174 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number 2
3-genus 4
Bridge index 2
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a174/ThurstonBennequinNumber
Hyperbolic Volume 12.1343
A-Polynomial See Data:K11a174/A-polynomial

[edit Notes for K11a174's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a174's four dimensional invariants]

Polynomial invariants

Alexander polynomial -t^4+5 t^3-11 t^2+15 t-15+15 t^{-1} -11 t^{-2} +5 t^{-3} - t^{-4}
Conway polynomial -z^8-3 z^6-z^4+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 79, -2 }
Jones polynomial -q^4+3 q^3-5 q^2+8 q-10+12 q^{-1} -12 q^{-2} +11 q^{-3} -8 q^{-4} +5 q^{-5} -3 q^{-6} + q^{-7}
HOMFLY-PT polynomial (db, data sources) -a^2 z^8+a^4 z^6-6 a^2 z^6+2 z^6+4 a^4 z^4-13 a^2 z^4-z^4 a^{-2} +9 z^4+4 a^4 z^2-12 a^2 z^2-3 z^2 a^{-2} +11 z^2+a^4-3 a^2- a^{-2} +4
Kauffman polynomial (db, data sources) a^2 z^{10}+z^{10}+3 a^3 z^9+6 a z^9+3 z^9 a^{-1} +4 a^4 z^8+5 a^2 z^8+3 z^8 a^{-2} +4 z^8+4 a^5 z^7-4 a^3 z^7-19 a z^7-10 z^7 a^{-1} +z^7 a^{-3} +4 a^6 z^6-5 a^4 z^6-24 a^2 z^6-13 z^6 a^{-2} -28 z^6+3 a^7 z^5-2 a^5 z^5+a^3 z^5+15 a z^5+5 z^5 a^{-1} -4 z^5 a^{-3} +a^8 z^4-4 a^6 z^4+3 a^4 z^4+31 a^2 z^4+16 z^4 a^{-2} +39 z^4-4 a^7 z^3-3 a^5 z^3-a^3 z^3-2 a z^3+4 z^3 a^{-1} +4 z^3 a^{-3} -a^8 z^2+a^6 z^2-2 a^4 z^2-17 a^2 z^2-7 z^2 a^{-2} -20 z^2+a^7 z+2 a^5 z+a^3 z-a z-2 z a^{-1} -z a^{-3} +a^4+3 a^2+ a^{-2} +4
The A2 invariant q^{20}-q^{18}+q^{16}-q^{14}-q^{12}+q^{10}-2 q^8+3 q^6-q^4+q^2+1- q^{-2} +2 q^{-4} + q^{-8} - q^{-12}
The G2 invariant q^{114}-2 q^{112}+4 q^{110}-6 q^{108}+4 q^{106}-2 q^{104}-4 q^{102}+12 q^{100}-17 q^{98}+22 q^{96}-21 q^{94}+10 q^{92}+5 q^{90}-22 q^{88}+36 q^{86}-43 q^{84}+41 q^{82}-27 q^{80}+7 q^{78}+20 q^{76}-40 q^{74}+56 q^{72}-55 q^{70}+45 q^{68}-29 q^{66}-q^{64}+29 q^{62}-52 q^{60}+63 q^{58}-55 q^{56}+33 q^{54}-4 q^{52}-30 q^{50}+47 q^{48}-45 q^{46}+21 q^{44}+11 q^{42}-42 q^{40}+48 q^{38}-24 q^{36}-19 q^{34}+67 q^{32}-95 q^{30}+89 q^{28}-45 q^{26}-24 q^{24}+92 q^{22}-133 q^{20}+133 q^{18}-86 q^{16}+15 q^{14}+58 q^{12}-105 q^{10}+117 q^8-88 q^6+32 q^4+25 q^2-67+72 q^{-2} -39 q^{-4} -7 q^{-6} +56 q^{-8} -76 q^{-10} +62 q^{-12} -15 q^{-14} -46 q^{-16} +98 q^{-18} -115 q^{-20} +92 q^{-22} -34 q^{-24} -32 q^{-26} +86 q^{-28} -105 q^{-30} +93 q^{-32} -53 q^{-34} +4 q^{-36} +32 q^{-38} -53 q^{-40} +49 q^{-42} -34 q^{-44} +16 q^{-46} + q^{-48} -10 q^{-50} +10 q^{-52} -9 q^{-54} +5 q^{-56} -2 q^{-58} + q^{-60}

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

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

Vassiliev invariants

V2 and V3: (0, 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
0 8 0 0 8 0 -\frac{208}{3} -\frac{160}{3} -24 0 32 0 0 208 \frac{536}{3} -\frac{104}{3} \frac{112}{3} -64

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 K11a174. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-6-5-4-3-2-1012345χ
9           1-1
7          2 2
5         31 -2
3        52  3
1       53   -2
-1      75    2
-3     66     0
-5    56      -1
-7   36       3
-9  25        -3
-11 13         2
-13 2          -2
-151           1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-3 i=-1
r=-6 {\mathbb Z}
r=-5 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-3 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-2 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-1 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=0 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{7}
r=1 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=3 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=4 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=5 {\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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K11a173.gif

K11a173

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K11a175