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

Visit K11n14 at Knotilus!

Knot presentations

Planar diagram presentation X4251 X8394 X10,6,11,5 X7,16,8,17 X2,9,3,10 X18,11,19,12 X20,13,21,14 X15,6,16,7 X22,17,1,18 X12,19,13,20 X14,21,15,22
Gauss code 1, -5, 2, -1, 3, 8, -4, -2, 5, -3, 6, -10, 7, -11, -8, 4, 9, -6, 10, -7, 11, -9
Dowker-Thistlethwaite code 4 8 10 -16 2 18 20 -6 22 12 14
A Braid Representative
A Morse Link Presentation K11n14 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number 2
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11n14/ThurstonBennequinNumber
Hyperbolic Volume 10.8397
A-Polynomial See Data:K11n14/A-polynomial

[edit Notes for K11n14's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 3
Rasmussen s-Invariant 4

[edit Notes for K11n14's four dimensional invariants]

Polynomial invariants

Alexander polynomial -t^3+6 t^2-10 t+11-10 t^{-1} +6 t^{-2} - t^{-3}
Conway polynomial -z^6+5 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 45, -4 }
Jones polynomial 2 q^{-2} -3 q^{-3} +5 q^{-4} -7 q^{-5} +8 q^{-6} -7 q^{-7} +6 q^{-8} -4 q^{-9} +2 q^{-10} - q^{-11}
HOMFLY-PT polynomial (db, data sources) -z^2 a^{10}-2 a^{10}+2 z^4 a^8+6 z^2 a^8+4 a^8-z^6 a^6-4 z^4 a^6-6 z^2 a^6-4 a^6+2 z^4 a^4+6 z^2 a^4+3 a^4
Kauffman polynomial (db, data sources) z^5 a^{13}-3 z^3 a^{13}+2 z a^{13}+2 z^6 a^{12}-5 z^4 a^{12}+2 z^2 a^{12}+2 z^7 a^{11}-3 z^5 a^{11}-z^3 a^{11}+2 z^8 a^{10}-5 z^6 a^{10}+8 z^4 a^{10}-8 z^2 a^{10}+2 a^{10}+z^9 a^9-z^7 a^9+3 z^3 a^9-z a^9+4 z^8 a^8-15 z^6 a^8+27 z^4 a^8-17 z^2 a^8+4 a^8+z^9 a^7-2 z^7 a^7+3 z^5 a^7+z a^7+2 z^8 a^6-8 z^6 a^6+17 z^4 a^6-15 z^2 a^6+4 a^6+z^7 a^5-z^5 a^5-z^3 a^5+3 z^4 a^4-8 z^2 a^4+3 a^4
The A2 invariant -q^{34}-q^{32}-q^{28}+2 q^{26}+q^{24}+q^{20}-2 q^{18}+q^{16}-q^{14}+2 q^{10}+2 q^6
The G2 invariant Data:K11n14/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11n121,}

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

Vassiliev invariants

V2 and V3: (5, -13)
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 -104 200 \frac{1894}{3} \frac{290}{3} -2080 -\frac{12560}{3} -\frac{2240}{3} -552 \frac{4000}{3} 5408 \frac{37880}{3} \frac{5800}{3} \frac{170431}{6} 814 \frac{99734}{9} \frac{3205}{18} \frac{8479}{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 K11n14. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
-3         22
-5        21-1
-7       31 2
-9      42  -2
-11     43   1
-13    34    1
-15   34     -1
-17  13      2
-19 13       -2
-21 1        1
-231         -1
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-5 i=-3
r=-9 {\mathbb Z}
r=-8 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-7 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-6 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-5 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-4 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-2 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-1 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=0 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}^{2}

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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