K11a79

K11a78

K11a80

(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

Three dimensional invariants

Symmetry type Chiral Unknotting number [math]\displaystyle{ \{1,2\} }[/math] 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 [math]\displaystyle{ 4 }[/math] Rasmussen s-Invariant 2

[edit Notes for K11a79's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ -t^4+6 t^3-17 t^2+30 t-35+30 t^{-1} -17 t^{-2} +6 t^{-3} - t^{-4} }[/math]
Conway polynomial	[math]\displaystyle{ -z^8-2 z^6-z^4+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 143, -2 }
Jones polynomial	[math]\displaystyle{ 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} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ -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 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ 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 }[/math]
The A2 invariant	[math]\displaystyle{ -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} }[/math]
The G2 invariant	[math]\displaystyle{ 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} }[/math]

Further Quantum Invariants

K11a79 Quantum Invariants.

Computer Talk

The above data is available with the Mathematica package KnotTheory`, as shown in the (simulated) Mathematica session below. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting. This Mathematica session is also available (albeit only for the knot 5_2) as the notebook PolynomialInvariantsSession.nb.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of August 31, 2006, 11:25:27.5625. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["K11a79"];

In[4]:=

Alexander[K][t]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]=

[math]\displaystyle{ -t^4+6 t^3-17 t^2+30 t-35+30 t^{-1} -17 t^{-2} +6 t^{-3} - t^{-4} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ -z^8-2 z^6-z^4+1 }[/math]

In[6]:=

Alexander[K, 2][t]

KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.

Out[6]=

[math]\displaystyle{ \{1\} }[/math]

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 143, -2 }

In[8]:=

Jones[K][q]

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[8]=

[math]\displaystyle{ 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} }[/math]

In[9]:=

HOMFLYPT[K][a, z]

KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.

Out[9]=

[math]\displaystyle{ -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 }[/math]

In[10]:=

Kauffman[K][a, z]

KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.

Out[10]=

[math]\displaystyle{ 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 }[/math]

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a255,}

Same Jones Polynomial (up to mirroring, [math]\displaystyle{ q\leftrightarrow q^{-1} }[/math]): {K11a255,}

Computer Talk

The above data is available with the Mathematica package KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of May 31, 2006, 14:15:20.091. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["K11a79"];

In[4]:=

{A = Alexander[K][t], J = Jones[K][q]}

KnotTheory::loading: Loading precomputed data in PD4Knots`.

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[4]=

{ [math]\displaystyle{ -t^4+6 t^3-17 t^2+30 t-35+30 t^{-1} -17 t^{-2} +6 t^{-3} - t^{-4} }[/math], [math]\displaystyle{ 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} }[/math] }

In[5]:=

DeleteCases[Select[AllKnots[], (A === Alexander[#][t]) &], K]

KnotTheory::loading: Loading precomputed data in DTCode4KnotsTo11`.

KnotTheory::credits: The GaussCode to PD conversion was written by Siddarth Sankaran at the University of Toronto in the summer of 2005.

Out[5]=

{K11a255,}

In[6]:=

DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]

KnotTheory::loading: Loading precomputed data in Jones4Knots11`.

Out[6]=

{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 [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ -8 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 32 }[/math] [math]\displaystyle{ 8 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ -\frac{80}{3} }[/math] [math]\displaystyle{ \frac{64}{3} }[/math] [math]\displaystyle{ -8 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 32 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 32 }[/math] [math]\displaystyle{ \frac{184}{3} }[/math] [math]\displaystyle{ -\frac{392}{3} }[/math] [math]\displaystyle{ \frac{64}{3} }[/math] [math]\displaystyle{ -48 }[/math]

V_2,1 through V_6,9 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 [math]\displaystyle{ t^rq^j }[/math] are shown, along with their alternating sums [math]\displaystyle{ \chi }[/math] (fixed [math]\displaystyle{ j }[/math], alternation over [math]\displaystyle{ r }[/math]). The squares with yellow highlighting are those on the "critical diagonals", where [math]\displaystyle{ j-2r=s+1 }[/math] or [math]\displaystyle{ j-2r=s-1 }[/math], where [math]\displaystyle{ s= }[/math]-2 is the signature of K11a79. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 7                       1 1 5                     3   -3 3                   6 1   5 1                 8 3     -5 -1               12 6       6 -3             12 9         -3 -5           11 11           0 -7         9 12             3 -9       6 11               -5 -11     3 9                 6 -13   1 6                   -5 -15   3                     3 -17 1                       -1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-3 }[/math] [math]\displaystyle{ i=-1 }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{12} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]

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). Back to the top.

K11a78 K11a80