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

Visit K11a5 at Knotilus!

Knot presentations

Planar diagram presentation X4251 X8394 X10,6,11,5 X14,7,15,8 X2,9,3,10 X20,12,21,11 X16,14,17,13 X6,15,7,16 X22,17,1,18 X12,20,13,19 X18,21,19,22
Gauss code 1, -5, 2, -1, 3, -8, 4, -2, 5, -3, 6, -10, 7, -4, 8, -7, 9, -11, 10, -6, 11, -9
Dowker-Thistlethwaite code 4 8 10 14 2 20 16 6 22 12 18
A Braid Representative
A Morse Link Presentation K11a5 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number \{1,2\}
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a5/ThurstonBennequinNumber
Hyperbolic Volume 15.5012
A-Polynomial See Data:K11a5/A-polynomial

[edit Notes for K11a5's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 1
Rasmussen s-Invariant 0

[edit Notes for K11a5's four dimensional invariants]

Polynomial invariants

Alexander polynomial -t^3+9 t^2-30 t+45-30 t^{-1} +9 t^{-2} - t^{-3}
Conway polynomial -z^6+3 z^4-3 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 125, 0 }
Jones polynomial -q^5+4 q^4-8 q^3+14 q^2-18 q+20-20 q^{-1} +17 q^{-2} -12 q^{-3} +7 q^{-4} -3 q^{-5} + q^{-6}
HOMFLY-PT polynomial (db, data sources) a^6-3 z^2 a^4-2 a^4+3 z^4 a^2+4 z^2 a^2+3 a^2-z^6-2 z^4-5 z^2-3+2 z^4 a^{-2} +2 z^2 a^{-2} +2 a^{-2} -z^2 a^{-4}
Kauffman polynomial (db, data sources) a^2 z^{10}+z^{10}+3 a^3 z^9+7 a z^9+4 z^9 a^{-1} +4 a^4 z^8+10 a^2 z^8+7 z^8 a^{-2} +13 z^8+3 a^5 z^7+3 a^3 z^7+a z^7+8 z^7 a^{-1} +7 z^7 a^{-3} +a^6 z^6-6 a^4 z^6-19 a^2 z^6-4 z^6 a^{-2} +4 z^6 a^{-4} -20 z^6-8 a^5 z^5-18 a^3 z^5-23 a z^5-24 z^5 a^{-1} -10 z^5 a^{-3} +z^5 a^{-5} -3 a^6 z^4-2 a^4 z^4+3 a^2 z^4-8 z^4 a^{-2} -6 z^4 a^{-4} +7 a^5 z^3+15 a^3 z^3+18 a z^3+16 z^3 a^{-1} +5 z^3 a^{-3} -z^3 a^{-5} +3 a^6 z^2+6 a^4 z^2+7 a^2 z^2+8 z^2 a^{-2} +3 z^2 a^{-4} +9 z^2-2 a^5 z-3 a^3 z-4 a z-4 z a^{-1} -z a^{-3} -a^6-2 a^4-3 a^2-2 a^{-2} -3
The A2 invariant Data:K11a5/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a5/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {K11a112,}

Vassiliev invariants

V2 and V3: (-3, 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
-12 16 72 50 14 -192 -\frac{704}{3} -\frac{224}{3} -16 -288 128 -600 -168 -\frac{1471}{10} -\frac{454}{15} -\frac{1022}{15} \frac{127}{6} -\frac{191}{10}

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=0 is the signature of K11a5. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
11           1-1
9          3 3
7         51 -4
5        93  6
3       95   -4
1      119    2
-1     1010     0
-3    710      -3
-5   510       5
-7  27        -5
-9 15         4
-11 2          -2
-131           1
Integral Khovanov Homology

(db, data source)

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