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

Visit K11a248 at Knotilus!

Knot presentations

Planar diagram presentation X6271 X8394 X10,6,11,5 X18,8,19,7 X16,9,17,10 X20,11,21,12 X22,13,1,14 X4,16,5,15 X2,17,3,18 X14,19,15,20 X12,21,13,22
Gauss code 1, -9, 2, -8, 3, -1, 4, -2, 5, -3, 6, -11, 7, -10, 8, -5, 9, -4, 10, -6, 11, -7
Dowker-Thistlethwaite code 6 8 10 18 16 20 22 4 2 14 12
A Braid Representative
A Morse Link Presentation K11a248 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number \{1,2\}
3-genus 4
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a248/ThurstonBennequinNumber
Hyperbolic Volume 17.7145
A-Polynomial See Data:K11a248/A-polynomial

[edit Notes for K11a248's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a248's four dimensional invariants]

Polynomial invariants

Alexander polynomial -t^4+6 t^3-18 t^2+34 t-41+34 t^{-1} -18 t^{-2} +6 t^{-3} - t^{-4}
Conway polynomial -z^8-2 z^6-2 z^4+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 159, -2 }
Jones polynomial q^3-4 q^2+9 q-15+22 q^{-1} -25 q^{-2} +26 q^{-3} -23 q^{-4} +17 q^{-5} -11 q^{-6} +5 q^{-7} - q^{-8}
HOMFLY-PT polynomial (db, data sources) -a^2 z^8+2 a^4 z^6-5 a^2 z^6+z^6-a^6 z^4+6 a^4 z^4-10 a^2 z^4+3 z^4-a^6 z^2+5 a^4 z^2-7 a^2 z^2+3 z^2+1
Kauffman polynomial (db, data sources) 3 a^4 z^{10}+3 a^2 z^{10}+9 a^5 z^9+17 a^3 z^9+8 a z^9+13 a^6 z^8+17 a^4 z^8+12 a^2 z^8+8 z^8+11 a^7 z^7-2 a^5 z^7-33 a^3 z^7-16 a z^7+4 z^7 a^{-1} +5 a^8 z^6-17 a^6 z^6-44 a^4 z^6-44 a^2 z^6+z^6 a^{-2} -21 z^6+a^9 z^5-15 a^7 z^5-14 a^5 z^5+21 a^3 z^5+10 a z^5-9 z^5 a^{-1} -4 a^8 z^4+4 a^6 z^4+32 a^4 z^4+45 a^2 z^4-2 z^4 a^{-2} +19 z^4+4 a^7 z^3+3 a^5 z^3-7 a^3 z^3-2 a z^3+4 z^3 a^{-1} -a^6 z^2-9 a^4 z^2-15 a^2 z^2-7 z^2+a^7 z+3 a^5 z+3 a^3 z+a z+1
The A2 invariant -q^{24}+2 q^{22}-2 q^{18}+4 q^{16}-5 q^{14}+q^{12}-2 q^8+6 q^6-4 q^4+5 q^2-1-2 q^{-2} +3 q^{-4} -2 q^{-6} + q^{-8}
The G2 invariant Data:K11a248/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a71,}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {K11a71,}

Vassiliev invariants

V2 and V3: (0, 0)
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 0 0 16 16 0 -64 32 -96 0 0 0 0 136 -\frac{832}{3} \frac{896}{3} \frac{344}{3} 40

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 K11a248. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
7           11
5          3 -3
3         61 5
1        93  -6
-1       136   7
-3      1310    -3
-5     1312     1
-7    1013      3
-9   713       -6
-11  410        6
-13 17         -6
-15 4          4
-171           -1
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-3 i=-1
r=-7 {\mathbb Z}
r=-6 {\mathbb Z}^{4}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-5 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-4 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=-3 {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=-2 {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{13} {\mathbb Z}^{13}
r=-1 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{13} {\mathbb Z}^{13}
r=0 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{13}
r=1 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=3 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=4 {\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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