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

Visit K11a53 at Knotilus!

Knot presentations

Planar diagram presentation X4251 X8394 X14,6,15,5 X16,7,17,8 X2,9,3,10 X18,11,19,12 X20,13,21,14 X22,16,1,15 X10,17,11,18 X12,19,13,20 X6,21,7,22
Gauss code 1, -5, 2, -1, 3, -11, 4, -2, 5, -9, 6, -10, 7, -3, 8, -4, 9, -6, 10, -7, 11, -8
Dowker-Thistlethwaite code 4 8 14 16 2 18 20 22 10 12 6
A Braid Representative
A Morse Link Presentation K11a53 ML.gif

Three dimensional invariants

Symmetry type Chiral
Unknotting number \{2,3\}
3-genus 4
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a53/ThurstonBennequinNumber
Hyperbolic Volume 14.416
A-Polynomial See Data:K11a53/A-polynomial

[edit Notes for K11a53'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 K11a53's four dimensional invariants]

Polynomial invariants

Alexander polynomial t^4-6 t^3+14 t^2-18 t+19-18 t^{-1} +14 t^{-2} -6 t^{-3} + t^{-4}
Conway polynomial z^8+2 z^6-2 z^4+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 97, -4 }
Jones polynomial -q+4-6 q^{-1} +10 q^{-2} -13 q^{-3} +15 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+a^8-2 z^6 a^6-8 z^4 a^6-8 z^2 a^6-2 a^6+z^8 a^4+5 z^6 a^4+8 z^4 a^4+5 z^2 a^4-z^6 a^2-3 z^4 a^2+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}+z a^{11}+5 z^6 a^{10}-5 z^4 a^{10}+2 z^2 a^{10}+6 z^7 a^9-6 z^5 a^9+z^3 a^9+z a^9+6 z^8 a^8-8 z^6 a^8+4 z^4 a^8-3 z^2 a^8+a^8+5 z^9 a^7-9 z^7 a^7+7 z^5 a^7-6 z^3 a^7+z a^7+2 z^{10} a^6+4 z^8 a^6-26 z^6 a^6+32 z^4 a^6-16 z^2 a^6+2 a^6+10 z^9 a^5-35 z^7 a^5+38 z^5 a^5-17 z^3 a^5+3 z a^5+2 z^{10} a^4+2 z^8 a^4-29 z^6 a^4+38 z^4 a^4-12 z^2 a^4+5 z^9 a^3-19 z^7 a^3+19 z^5 a^3-6 z^3 a^3+2 z a^3+4 z^8 a^2-16 z^6 a^2+16 z^4 a^2-2 z^2 a^2-2 a^2+z^7 a-3 z^5 a+z^3 a
The A2 invariant q^{30}-q^{26}+q^{24}-2 q^{22}+2 q^{20}-q^{18}-q^{16}+q^{14}-4 q^{12}+3 q^{10}-q^8+2 q^6+2 q^4+2- q^{-2}
The G2 invariant q^{162}-2 q^{160}+4 q^{158}-6 q^{156}+5 q^{154}-4 q^{152}-2 q^{150}+11 q^{148}-20 q^{146}+28 q^{144}-30 q^{142}+22 q^{140}-4 q^{138}-21 q^{136}+50 q^{134}-68 q^{132}+71 q^{130}-56 q^{128}+20 q^{126}+22 q^{124}-60 q^{122}+91 q^{120}-96 q^{118}+87 q^{116}-61 q^{114}+18 q^{112}+33 q^{110}-87 q^{108}+125 q^{106}-133 q^{104}+99 q^{102}-29 q^{100}-57 q^{98}+125 q^{96}-140 q^{94}+94 q^{92}-100 q^{88}+143 q^{86}-110 q^{84}+9 q^{82}+119 q^{80}-205 q^{78}+213 q^{76}-129 q^{74}-20 q^{72}+170 q^{70}-272 q^{68}+278 q^{66}-195 q^{64}+47 q^{62}+107 q^{60}-213 q^{58}+249 q^{56}-202 q^{54}+87 q^{52}+43 q^{50}-151 q^{48}+181 q^{46}-129 q^{44}+16 q^{42}+113 q^{40}-183 q^{38}+167 q^{36}-68 q^{34}-73 q^{32}+199 q^{30}-249 q^{28}+203 q^{26}-79 q^{24}-66 q^{22}+181 q^{20}-216 q^{18}+181 q^{16}-90 q^{14}-4 q^{12}+72 q^{10}-101 q^8+91 q^6-55 q^4+21 q^2+7-18 q^{-2} +17 q^{-4} -14 q^{-6} +7 q^{-8} -3 q^{-10} + q^{-12}

"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, 2)
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 16 0 -48 16 0 -\frac{32}{3} -\frac{224}{3} -80 0 128 0 0 680 -\frac{608}{3} \frac{2816}{3} \frac{152}{3} 104

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 K11a53. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
3           1-1
1          3 3
-1         31 -2
-3        73  4
-5       74   -3
-7      86    2
-9     77     0
-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}^{7}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=-1 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=0 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{7}
r=1 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=2 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
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

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See/edit the Hoste-Thistlethwaite Knot Page master template (intermediate).

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