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

Visit K11a27 at Knotilus!

Knot presentations

Planar diagram presentation X4251 X8394 X12,6,13,5 X14,7,15,8 X2,9,3,10 X18,11,19,12 X22,14,1,13 X20,15,21,16 X10,17,11,18 X16,19,17,20 X6,21,7,22
Gauss code 1, -5, 2, -1, 3, -11, 4, -2, 5, -9, 6, -3, 7, -4, 8, -10, 9, -6, 10, -8, 11, -7
Dowker-Thistlethwaite code 4 8 12 14 2 18 22 20 10 16 6
A Braid Representative
A Morse Link Presentation K11a27 ML.gif

Three dimensional invariants

Symmetry type Chiral
Unknotting number \{1,2\}
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a27/ThurstonBennequinNumber
Hyperbolic Volume 17.0192
A-Polynomial See Data:K11a27/A-polynomial

[edit Notes for K11a27's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 3
Rasmussen s-Invariant 2

[edit Notes for K11a27's four dimensional invariants]

Polynomial invariants

Alexander polynomial 2 t^3-13 t^2+34 t-45+34 t^{-1} -13 t^{-2} +2 t^{-3}
Conway polynomial 2 z^6-z^4+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 143, -2 }
Jones polynomial -q^2+4 q-8+15 q^{-1} -20 q^{-2} +23 q^{-3} -23 q^{-4} +20 q^{-5} -15 q^{-6} +9 q^{-7} -4 q^{-8} + q^{-9}
HOMFLY-PT polynomial (db, data sources) z^2 a^8-2 z^4 a^6-z^2 a^6+a^6+z^6 a^4-3 z^2 a^4-3 a^4+z^6 a^2+2 z^4 a^2+4 z^2 a^2+3 a^2-z^4-z^2
Kauffman polynomial (db, data sources) z^6 a^{10}-2 z^4 a^{10}+z^2 a^{10}+4 z^7 a^9-9 z^5 a^9+6 z^3 a^9-z a^9+7 z^8 a^8-15 z^6 a^8+9 z^4 a^8-2 z^2 a^8+6 z^9 a^7-5 z^7 a^7-10 z^5 a^7+9 z^3 a^7-2 z a^7+2 z^{10} a^6+13 z^8 a^6-35 z^6 a^6+22 z^4 a^6-3 z^2 a^6-a^6+12 z^9 a^5-14 z^7 a^5-7 z^5 a^5+10 z^3 a^5-2 z a^5+2 z^{10} a^4+14 z^8 a^4-29 z^6 a^4+11 z^4 a^4+6 z^2 a^4-3 a^4+6 z^9 a^3+2 z^7 a^3-16 z^5 a^3+12 z^3 a^3-2 z a^3+8 z^8 a^2-6 z^6 a^2-6 z^4 a^2+9 z^2 a^2-3 a^2+7 z^7 a-9 z^5 a+4 z^3 a-z a+4 z^6-6 z^4+3 z^2+z^5 a^{-1} -z^3 a^{-1}
The A2 invariant Data:K11a27/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a27/QuantumInvariant/G2/1,0

"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 -16 8 0 -\frac{112}{3} -\frac{160}{3} -24 0 32 0 0 296 -\frac{152}{3} \frac{1096}{3} -\frac{8}{3} 56

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 K11a27. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
5           1-1
3          3 3
1         51 -4
-1        103  7
-3       116   -5
-5      129    3
-7     1111     0
-9    912      -3
-11   611       5
-13  39        -6
-15 16         5
-17 3          -3
-191           1
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-3 i=-1
r=-8 {\mathbb Z}
r=-7 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-6 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-5 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=-4 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=-3 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=-2 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r=-1 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=0 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{10}
r=1 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
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

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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