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

Visit K11n97 at Knotilus!

Knot presentations

Planar diagram presentation X4251 X10,3,11,4 X14,6,15,5 X12,8,13,7 X9,19,10,18 X2,11,3,12 X6,14,7,13 X15,20,16,21 X17,22,18,1 X19,9,20,8 X21,16,22,17
Gauss code 1, -6, 2, -1, 3, -7, 4, 10, -5, -2, 6, -4, 7, -3, -8, 11, -9, 5, -10, 8, -11, 9
Dowker-Thistlethwaite code 4 10 14 12 -18 2 6 -20 -22 -8 -16
A Braid Representative
A Morse Link Presentation K11n97 ML.gif

Three dimensional invariants

Symmetry type Chiral
Unknotting number \{1,2\}
3-genus 2
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11n97/ThurstonBennequinNumber
Hyperbolic Volume 10.183
A-Polynomial See Data:K11n97/A-polynomial

[edit Notes for K11n97'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 K11n97's four dimensional invariants]

Polynomial invariants

Alexander polynomial -2 t+5-2 t^{-1}
Conway polynomial 1-2 z^2
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 9, 0 }
Jones polynomial q^5-q^4+q^3-q^2-q+2-2 q^{-1} +3 q^{-2} -2 q^{-3} +2 q^{-4} - q^{-5}
HOMFLY-PT polynomial (db, data sources) -z^2 a^4-a^4+z^4 a^2+3 z^2 a^2+3 a^2-z^2-z^4 a^{-2} -4 z^2 a^{-2} -3 a^{-2} +z^2 a^{-4} +2 a^{-4}
Kauffman polynomial (db, data sources) z^9 a^{-1} +z^9 a^{-3} +2 z^8 a^{-2} +z^8 a^{-4} +z^8+a^3 z^7-8 z^7 a^{-1} -7 z^7 a^{-3} +2 a^4 z^6+2 a^2 z^6-15 z^6 a^{-2} -7 z^6 a^{-4} -8 z^6+a^5 z^5-2 a^3 z^5+a z^5+19 z^5 a^{-1} +15 z^5 a^{-3} -7 a^4 z^4-7 a^2 z^4+32 z^4 a^{-2} +15 z^4 a^{-4} +17 z^4-3 a^5 z^3-2 a^3 z^3-4 a z^3-18 z^3 a^{-1} -13 z^3 a^{-3} +5 a^4 z^2+7 a^2 z^2-22 z^2 a^{-2} -11 z^2 a^{-4} -9 z^2+a^5 z+2 a^3 z+4 a z+8 z a^{-1} +5 z a^{-3} -a^4-3 a^2+3 a^{-2} +2 a^{-4}
The A2 invariant Data:K11n97/QuantumInvariant/A2/1,0
The G2 invariant Data:K11n97/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {6_1, 9_46, K11n67, K11n139,}

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

Vassiliev invariants

V2 and V3: (-2, -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
-8 -24 32 \frac{116}{3} \frac{52}{3} 192 336 96 72 -\frac{256}{3} 288 -\frac{928}{3} -\frac{416}{3} \frac{4409}{15} -\frac{796}{15} \frac{2876}{45} \frac{775}{9} -\frac{151}{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 K11n97. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
11           11
9            0
7         11 0
5       11   0
3      1 1   -2
1     221    1
-1    22      0
-3   111      1
-5  12        1
-7 11         0
-9 1          1
-111           -1
Integral Khovanov Homology

(db, data source)

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