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

Visit K11a194 at Knotilus!

Knot presentations

Planar diagram presentation X4251 X12,3,13,4 X14,6,15,5 X20,7,21,8 X16,9,17,10 X18,11,19,12 X2,13,3,14 X22,16,1,15 X10,17,11,18 X6,19,7,20 X8,21,9,22
Gauss code 1, -7, 2, -1, 3, -10, 4, -11, 5, -9, 6, -2, 7, -3, 8, -5, 9, -6, 10, -4, 11, -8
Dowker-Thistlethwaite code 4 12 14 20 16 18 2 22 10 6 8
A Braid Representative
A Morse Link Presentation K11a194 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number 2
3-genus 4
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a194/ThurstonBennequinNumber
Hyperbolic Volume 13.7968
A-Polynomial See Data:K11a194/A-polynomial

[edit Notes for K11a194'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 K11a194's four dimensional invariants]

Polynomial invariants

Alexander polynomial t^4-5 t^3+12 t^2-18 t+21-18 t^{-1} +12 t^{-2} -5 t^{-3} + t^{-4}
Conway polynomial z^8+3 z^6+2 z^4+z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 93, -4 }
Jones polynomial -q+3-5 q^{-1} +10 q^{-2} -12 q^{-3} +14 q^{-4} -15 q^{-5} +13 q^{-6} -10 q^{-7} +6 q^{-8} -3 q^{-9} + q^{-10}
HOMFLY-PT polynomial (db, data sources) z^4 a^8+3 z^2 a^8+2 a^8-2 z^6 a^6-9 z^4 a^6-13 z^2 a^6-6 a^6+z^8 a^4+6 z^6 a^4+14 z^4 a^4+15 z^2 a^4+5 a^4-z^6 a^2-4 z^4 a^2-4 z^2 a^2
Kauffman polynomial (db, data sources) z^4 a^{12}-z^2 a^{12}+3 z^5 a^{11}-3 z^3 a^{11}+5 z^6 a^{10}-5 z^4 a^{10}+z^2 a^{10}+7 z^7 a^9-12 z^5 a^9+11 z^3 a^9-4 z a^9+7 z^8 a^8-14 z^6 a^8+13 z^4 a^8-5 z^2 a^8+2 a^8+4 z^9 a^7-z^7 a^7-17 z^5 a^7+23 z^3 a^7-8 z a^7+z^{10} a^6+10 z^8 a^6-39 z^6 a^6+46 z^4 a^6-25 z^2 a^6+6 a^6+7 z^9 a^5-18 z^7 a^5+7 z^5 a^5+5 z^3 a^5-3 z a^5+z^{10} a^4+6 z^8 a^4-33 z^6 a^4+45 z^4 a^4-27 z^2 a^4+5 a^4+3 z^9 a^3-9 z^7 a^3+5 z^5 a^3+z a^3+3 z^8 a^2-13 z^6 a^2+18 z^4 a^2-9 z^2 a^2+z^7 a-4 z^5 a+4 z^3 a
The A2 invariant q^{30}+q^{24}-3 q^{22}+q^{20}-2 q^{18}-q^{16}+q^{14}-3 q^{12}+4 q^{10}+3 q^6+2 q^4-q^2+1- q^{-2}
The G2 invariant q^{162}-2 q^{160}+4 q^{158}-6 q^{156}+5 q^{154}-3 q^{152}-2 q^{150}+10 q^{148}-17 q^{146}+24 q^{144}-27 q^{142}+19 q^{140}-7 q^{138}-13 q^{136}+36 q^{134}-55 q^{132}+68 q^{130}-65 q^{128}+41 q^{126}+q^{124}-47 q^{122}+96 q^{120}-124 q^{118}+125 q^{116}-90 q^{114}+17 q^{112}+68 q^{110}-135 q^{108}+168 q^{106}-138 q^{104}+66 q^{102}+28 q^{100}-112 q^{98}+143 q^{96}-110 q^{94}+17 q^{92}+82 q^{90}-147 q^{88}+138 q^{86}-54 q^{84}-71 q^{82}+185 q^{80}-241 q^{78}+202 q^{76}-92 q^{74}-74 q^{72}+215 q^{70}-286 q^{68}+263 q^{66}-150 q^{64}-5 q^{62}+143 q^{60}-221 q^{58}+215 q^{56}-136 q^{54}+6 q^{52}+107 q^{50}-157 q^{48}+135 q^{46}-36 q^{44}-76 q^{42}+165 q^{40}-176 q^{38}+112 q^{36}-q^{34}-123 q^{32}+211 q^{30}-210 q^{28}+146 q^{26}-34 q^{24}-75 q^{22}+150 q^{20}-163 q^{18}+126 q^{16}-62 q^{14}-6 q^{12}+53 q^{10}-70 q^8+61 q^6-37 q^4+16 q^2+3-13 q^{-2} +11 q^{-4} -10 q^{-6} +5 q^{-8} -2 q^{-10} + q^{-12}

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a106, K11a346,}

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{226}{3} -\frac{38}{3} 32 \frac{944}{3} -\frac{64}{3} 104 \frac{32}{3} 32 -\frac{904}{3} -\frac{152}{3} -\frac{26129}{30} \frac{5738}{15} -\frac{24058}{45} -\frac{2191}{18} -\frac{2129}{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=-4 is the signature of K11a194. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
3           1-1
1          2 2
-1         31 -2
-3        72  5
-5       64   -2
-7      86    2
-9     76     -1
-11    68      -2
-13   47       3
-15  26        -4
-17 14         3
-19 2          -2
-211           1
Integral Khovanov Homology

(db, data source)

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