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

Visit K11a79 at Knotilus!

Knot presentations

Planar diagram presentation X4251 X10,3,11,4 X12,6,13,5 X14,7,15,8 X20,10,21,9 X2,11,3,12 X18,13,19,14 X8,15,9,16 X22,17,1,18 X6,20,7,19 X16,21,17,22
Gauss code 1, -6, 2, -1, 3, -10, 4, -8, 5, -2, 6, -3, 7, -4, 8, -11, 9, -7, 10, -5, 11, -9
Dowker-Thistlethwaite code 4 10 12 14 20 2 18 8 22 6 16
A Braid Representative
A Morse Link Presentation K11a79 ML.gif

Three dimensional invariants

Symmetry type Chiral
Unknotting number \{1,2\}
3-genus 4
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a79/ThurstonBennequinNumber
Hyperbolic Volume 16.8714
A-Polynomial See Data:K11a79/A-polynomial

[edit Notes for K11a79'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 K11a79's four dimensional invariants]

Polynomial invariants

Alexander polynomial -t^4+6 t^3-17 t^2+30 t-35+30 t^{-1} -17 t^{-2} +6 t^{-3} - t^{-4}
Conway polynomial -z^8-2 z^6-z^4+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 143, -2 }
Jones polynomial q^3-4 q^2+9 q-14+20 q^{-1} -23 q^{-2} +23 q^{-3} -20 q^{-4} +15 q^{-5} -9 q^{-6} +4 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+7 a^4 z^4-10 a^2 z^4+3 z^4-2 a^6 z^2+8 a^4 z^2-9 a^2 z^2+3 z^2-a^6+3 a^4-3 a^2+2
Kauffman polynomial (db, data sources) 2 a^4 z^{10}+2 a^2 z^{10}+7 a^5 z^9+13 a^3 z^9+6 a z^9+10 a^6 z^8+17 a^4 z^8+14 a^2 z^8+7 z^8+8 a^7 z^7-2 a^5 z^7-20 a^3 z^7-6 a z^7+4 z^7 a^{-1} +4 a^8 z^6-16 a^6 z^6-48 a^4 z^6-45 a^2 z^6+z^6 a^{-2} -16 z^6+a^9 z^5-12 a^7 z^5-14 a^5 z^5-a^3 z^5-9 a z^5-9 z^5 a^{-1} -5 a^8 z^4+11 a^6 z^4+47 a^4 z^4+44 a^2 z^4-2 z^4 a^{-2} +11 z^4-a^9 z^3+6 a^7 z^3+16 a^5 z^3+14 a^3 z^3+10 a z^3+5 z^3 a^{-1} +a^8 z^2-5 a^6 z^2-19 a^4 z^2-19 a^2 z^2+z^2 a^{-2} -5 z^2-2 a^7 z-5 a^5 z-5 a^3 z-3 a z-z a^{-1} +a^6+3 a^4+3 a^2+2
The A2 invariant -q^{24}+q^{22}-2 q^{18}+4 q^{16}-3 q^{14}+2 q^{12}+q^{10}-3 q^8+4 q^6-5 q^4+4 q^2- q^{-2} +3 q^{-4} -2 q^{-6} + q^{-8}
The G2 invariant q^{128}-3 q^{126}+7 q^{124}-13 q^{122}+16 q^{120}-16 q^{118}+7 q^{116}+16 q^{114}-46 q^{112}+84 q^{110}-113 q^{108}+112 q^{106}-74 q^{104}-16 q^{102}+148 q^{100}-283 q^{98}+379 q^{96}-381 q^{94}+242 q^{92}+16 q^{90}-342 q^{88}+633 q^{86}-760 q^{84}+651 q^{82}-302 q^{80}-187 q^{78}+631 q^{76}-861 q^{74}+773 q^{72}-393 q^{70}-127 q^{68}+559 q^{66}-710 q^{64}+511 q^{62}-33 q^{60}-488 q^{58}+816 q^{56}-780 q^{54}+361 q^{52}+276 q^{50}-887 q^{48}+1228 q^{46}-1134 q^{44}+638 q^{42}+99 q^{40}-808 q^{38}+1233 q^{36}-1234 q^{34}+817 q^{32}-165 q^{30}-482 q^{28}+876 q^{26}-877 q^{24}+522 q^{22}+30 q^{20}-521 q^{18}+724 q^{16}-570 q^{14}+113 q^{12}+425 q^{10}-804 q^8+875 q^6-594 q^4+88 q^2+442-796 q^{-2} +861 q^{-4} -643 q^{-6} +256 q^{-8} +139 q^{-10} -416 q^{-12} +513 q^{-14} -437 q^{-16} +268 q^{-18} -68 q^{-20} -83 q^{-22} +155 q^{-24} -163 q^{-26} +121 q^{-28} -65 q^{-30} +20 q^{-32} +12 q^{-34} -24 q^{-36} +22 q^{-38} -16 q^{-40} +8 q^{-42} -3 q^{-44} + q^{-46}

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a255,}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {K11a255,}

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 32 8 0 -\frac{80}{3} \frac{64}{3} -8 0 32 0 0 32 \frac{184}{3} -\frac{392}{3} \frac{64}{3} -48

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 K11a79. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
7           11
5          3 -3
3         61 5
1        83  -5
-1       126   6
-3      129    -3
-5     1111     0
-7    912      3
-9   611       -5
-11  39        6
-13 16         -5
-15 3          3
-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}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-5 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-4 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=-3 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=-2 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=-1 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r=0 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{12}
r=1 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
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

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