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

Visit K11a15 at Knotilus!

Knot presentations

Planar diagram presentation X4251 X8493 X12,5,13,6 X2837 X14,9,15,10 X18,12,19,11 X6,13,7,14 X20,15,21,16 X22,17,1,18 X10,20,11,19 X16,21,17,22
Gauss code 1, -4, 2, -1, 3, -7, 4, -2, 5, -10, 6, -3, 7, -5, 8, -11, 9, -6, 10, -8, 11, -9
Dowker-Thistlethwaite code 4 8 12 2 14 18 6 20 22 10 16
A Braid Representative
A Morse Link Presentation K11a15 ML.gif

Three dimensional invariants

Symmetry type Chiral
Unknotting number \{1,2\}
3-genus 4
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a15/ThurstonBennequinNumber
Hyperbolic Volume 15.1288
A-Polynomial See Data:K11a15/A-polynomial

[edit Notes for K11a15'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 K11a15's four dimensional invariants]

Polynomial invariants

Alexander polynomial -t^4+5 t^3-13 t^2+22 t-25+22 t^{-1} -13 t^{-2} +5 t^{-3} - t^{-4}
Conway polynomial -z^8-3 z^6-3 z^4-z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 107, -2 }
Jones polynomial -q^4+3 q^3-6 q^2+11 q-14+17 q^{-1} -17 q^{-2} +15 q^{-3} -12 q^{-4} +7 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^4-15 a^2 z^4-z^4 a^{-2} +9 z^4+6 a^4 z^2-18 a^2 z^2-3 z^2 a^{-2} +14 z^2+3 a^4-8 a^2-2 a^{-2} +8
Kauffman polynomial (db, data sources) a^2 z^{10}+z^{10}+5 a^3 z^9+8 a z^9+3 z^9 a^{-1} +9 a^4 z^8+15 a^2 z^8+3 z^8 a^{-2} +9 z^8+9 a^5 z^7+a^3 z^7-15 a z^7-6 z^7 a^{-1} +z^7 a^{-3} +6 a^6 z^6-16 a^4 z^6-52 a^2 z^6-12 z^6 a^{-2} -42 z^6+3 a^7 z^5-15 a^5 z^5-25 a^3 z^5-9 a z^5-6 z^5 a^{-1} -4 z^5 a^{-3} +a^8 z^4-6 a^6 z^4+13 a^4 z^4+58 a^2 z^4+16 z^4 a^{-2} +54 z^4-2 a^7 z^3+14 a^5 z^3+29 a^3 z^3+23 a z^3+15 z^3 a^{-1} +5 z^3 a^{-3} -a^8 z^2+3 a^6 z^2-7 a^4 z^2-33 a^2 z^2-9 z^2 a^{-2} -31 z^2-5 a^5 z-11 a^3 z-10 a z-6 z a^{-1} -2 z a^{-3} +3 a^4+8 a^2+2 a^{-2} +8
The A2 invariant q^{20}-q^{18}+3 q^{16}-q^{14}-q^{12}+q^{10}-5 q^8+2 q^6-3 q^4+2 q^2+3+4 q^{-4} - q^{-6} - q^{-12}
The G2 invariant q^{114}-2 q^{112}+4 q^{110}-6 q^{108}+6 q^{106}-5 q^{104}+9 q^{100}-19 q^{98}+29 q^{96}-37 q^{94}+33 q^{92}-20 q^{90}-5 q^{88}+44 q^{86}-78 q^{84}+105 q^{82}-114 q^{80}+87 q^{78}-31 q^{76}-56 q^{74}+155 q^{72}-222 q^{70}+240 q^{68}-180 q^{66}+52 q^{64}+114 q^{62}-253 q^{60}+318 q^{58}-267 q^{56}+116 q^{54}+78 q^{52}-231 q^{50}+278 q^{48}-189 q^{46}+10 q^{44}+180 q^{42}-295 q^{40}+259 q^{38}-97 q^{36}-144 q^{34}+352 q^{32}-446 q^{30}+366 q^{28}-152 q^{26}-136 q^{24}+378 q^{22}-501 q^{20}+449 q^{18}-254 q^{16}-18 q^{14}+262 q^{12}-389 q^{10}+367 q^8-197 q^6-28 q^4+220 q^2-291+223 q^{-2} -32 q^{-4} -173 q^{-6} +318 q^{-8} -320 q^{-10} +188 q^{-12} +29 q^{-14} -235 q^{-16} +359 q^{-18} -346 q^{-20} +222 q^{-22} -39 q^{-24} -136 q^{-26} +236 q^{-28} -246 q^{-30} +181 q^{-32} -80 q^{-34} -17 q^{-36} +76 q^{-38} -97 q^{-40} +81 q^{-42} -48 q^{-44} +17 q^{-46} +5 q^{-48} -16 q^{-50} +14 q^{-52} -11 q^{-54} +6 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: (-1, 3)
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 24 8 \frac{34}{3} \frac{86}{3} -96 -272 -96 -104 -\frac{32}{3} 288 -\frac{136}{3} -\frac{344}{3} \frac{22289}{30} \frac{3622}{15} \frac{778}{45} \frac{2383}{18} -\frac{2671}{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 K11a15. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
9           1-1
7          2 2
5         41 -3
3        72  5
1       74   -3
-1      107    3
-3     88     0
-5    79      -2
-7   58       3
-9  27        -5
-11 15         4
-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}^{5}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-3 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-2 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=-1 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=0 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{10}
r=1 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=3 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
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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