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

Visit K11n39 at Knotilus!

Knot presentations

Planar diagram presentation X4251 X8493 X12,5,13,6 X2837 X9,18,10,19 X11,17,12,16 X6,13,7,14 X15,21,16,20 X17,22,18,1 X19,15,20,14 X21,10,22,11
Gauss code 1, -4, 2, -1, 3, -7, 4, -2, -5, 11, -6, -3, 7, 10, -8, 6, -9, 5, -10, 8, -11, 9
Dowker-Thistlethwaite code 4 8 12 2 -18 -16 6 -20 -22 -14 -10
A Braid Representative
A Morse Link Presentation K11n39 ML.gif

Three dimensional invariants

Symmetry type Chiral
Unknotting number 1
3-genus 2
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11n39/ThurstonBennequinNumber
Hyperbolic Volume 12.7511
A-Polynomial See Data:K11n39/A-polynomial

[edit Notes for K11n39's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11n39's four dimensional invariants]

Polynomial invariants

Alexander polynomial 2 t^2-6 t+9-6 t^{-1} +2 t^{-2}
Conway polynomial 2 z^4+2 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 25, 0 }
Jones polynomial q^6-3 q^5+4 q^4-5 q^3+5 q^2-4 q+4- q^{-1} + q^{-3} - q^{-4}
HOMFLY-PT polynomial (db, data sources) -z^6 a^{-2} +z^6-a^2 z^4-5 z^4 a^{-2} +z^4 a^{-4} +7 z^4-4 a^2 z^2-9 z^2 a^{-2} +2 z^2 a^{-4} +13 z^2-3 a^2-5 a^{-2} + a^{-4} +8
Kauffman polynomial (db, data sources) z^9 a^{-1} +z^9 a^{-3} +a^2 z^8+4 z^8 a^{-2} +3 z^8 a^{-4} +2 z^8+a^3 z^7+a z^7-3 z^7 a^{-1} +3 z^7 a^{-5} -7 a^2 z^6-20 z^6 a^{-2} -11 z^6 a^{-4} +z^6 a^{-6} -15 z^6-6 a^3 z^5-10 a z^5-7 z^5 a^{-1} -14 z^5 a^{-3} -11 z^5 a^{-5} +13 a^2 z^4+30 z^4 a^{-2} +8 z^4 a^{-4} -3 z^4 a^{-6} +32 z^4+9 a^3 z^3+20 a z^3+23 z^3 a^{-1} +20 z^3 a^{-3} +8 z^3 a^{-5} -10 a^2 z^2-18 z^2 a^{-2} -3 z^2 a^{-4} +z^2 a^{-6} -24 z^2-4 a^3 z-10 a z-12 z a^{-1} -8 z a^{-3} -2 z a^{-5} +3 a^2+5 a^{-2} + a^{-4} +8
The A2 invariant -q^{12}-q^{10}-q^8+q^6+3 q^2+3+ q^{-2} +2 q^{-4} -2 q^{-6} -2 q^{-10} - q^{-12} + q^{-14} - q^{-16} + q^{-18}
The G2 invariant Data:K11n39/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {8_8, 10_129, K11n45, K11n50, K11n132,}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {K11n45,}

Vassiliev invariants

V2 and V3: (2, -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
8 -8 32 \frac{76}{3} -\frac{4}{3} -64 -\frac{272}{3} \frac{64}{3} -40 \frac{256}{3} 32 \frac{608}{3} -\frac{32}{3} \frac{3751}{15} \frac{476}{15} \frac{1204}{45} \frac{329}{9} -\frac{89}{15}

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 K11n39. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
13           11
11          2 -2
9         21 1
7        32  -1
5      132   0
3      23    1
1    143     0
-1   113      3
-3   12       -1
-5 111        1
-7            0
-91           -1
Integral Khovanov Homology

(db, data source)

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