© | Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots:

K11n158 K11n160

Knotscape

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DrawMorseLink

PD Presentation:

X6271 X10,3,11,4 X12,6,13,5 X7,18,8,19 X9,20,10,21 X16,11,17,12 X22,13,1,14 X4,16,5,15 X2,17,3,18 X19,8,20,9 X14,21,15,22

Gauss Code:

{1, -9, 2, -8, 3, -1, -4, 10, -5, -2, 6, -3, 7, -11, 8, -6, 9, 4, -10, 5, 11, -7}

DT (Dowker-Thistlethwaite) Code:

6 10 12 -18 -20 16 22 4 2 -8 14

Alexander Polynomial:

t-3 - 6t-2 + 17t-1 - 23 + 17t - 6t2 + t3

Conway Polynomial:

1 + 2z2 + z6

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{71, -2}

Jones Polynomial:

q-9 - 4q-8 + 7q-7 - 10q-6 + 12q-5 - 12q-4 + 11q-3 - 8q-2 + 5q-1 - 1

Other knots (up to mirrors) with the same Jones Polynomial:

{...}

A2 (sl(3)) Invariant:

q-28 - q-26 - 2q-24 + 2q-22 - 2q-20 + q-18 + q-16 - 2q-14 + 2q-12 - 2q-10 + 3q-8 + q-6 - q-4 + 3q-2 - 1

HOMFLY-PT Polynomial:

a2 - a2z4 + a4 + 4a4z2 + 3a4z4 + a4z6 - a6 - 3a6z2 - 2a6z4 + a8z2

Kauffman Polynomial:

az3 - a2 - a2z2 + 5a2z4 + a3z3 + a3z5 + 2a3z7 + a4 - 6a4z2 + 12a4z4 - 8a4z6 + 4a4z8 - 2a5z + 6a5z3 - 9a5z5 + 2a5z7 + 2a5z9 + a6 - 5a6z2 + 13a6z4 - 21a6z6 + 9a6z8 - 2a7z + 13a7z3 - 21a7z5 + 4a7z7 + 2a7z9 + a8z2 + 4a8z4 - 12a8z6 + 5a8z8 + 7a9z3 - 11a9z5 + 4a9z7 + a10z2 - 2a10z4 + a10z6

V2 and V3, the type 2 and 3 Vassiliev invariants:

{2, -3}

Khovanov Homology: (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 11159. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 j = 1                   1 j = -1                 4   j = -3               5 2   j = -5             6 3     j = -7           6 5       j = -9         6 6         j = -11       4 6           j = -13     3 6             j = -15   1 4               j = -17   3                 j = -19 1

Computer Talk. The data above can be recomputed by Mathematica using the package KnotTheory`. Following setup, the sample Mathematica session below reproduces most of the above data (Mathematica system prompts in blue, human input in red, Mathematica output in black):

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	Show[DrawMorseLink[Knot[11, NonAlternating, 159]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 159]]
Out[3]=	PD[X[6, 2, 7, 1], X[10, 3, 11, 4], X[12, 6, 13, 5], X[7, 18, 8, 19], > X[9, 20, 10, 21], X[16, 11, 17, 12], X[22, 13, 1, 14], X[4, 16, 5, 15], > X[2, 17, 3, 18], X[19, 8, 20, 9], X[14, 21, 15, 22]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 159]]
Out[4]=	GaussCode[1, -9, 2, -8, 3, -1, -4, 10, -5, -2, 6, -3, 7, -11, 8, -6, 9, 4, -10, > 5, 11, -7]
In[5]:=	DTCode[Knot[11, NonAlternating, 159]]
Out[5]=	DTCode[6, 10, 12, -18, -20, 16, 22, 4, 2, -8, 14]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 159]][t]
Out[6]=	-3 6 17 2 3 -23 + t - -- + -- + 17 t - 6 t + t 2 t t
In[7]:=	Conway[Knot[11, NonAlternating, 159]][z]
Out[7]=	2 6 1 + 2 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, NonAlternating, 159]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 159]], KnotSignature[Knot[11, NonAlternating, 159]]}
Out[9]=	{71, -2}
In[10]:=	J=Jones[Knot[11, NonAlternating, 159]][q]
Out[10]=	-9 4 7 10 12 12 11 8 5 -1 + q - -- + -- - -- + -- - -- + -- - -- + - 8 7 6 5 4 3 2 q q q q q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, NonAlternating, 159]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 159]][q]
Out[12]=	-28 -26 2 2 2 -18 -16 2 2 2 3 -6 -1 + q - q - --- + --- - --- + q + q - --- + --- - --- + -- + q - 24 22 20 14 12 10 8 q q q q q q q -4 3 > q + -- 2 q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 159]][a, z]
Out[13]=	2 4 6 4 2 6 2 8 2 2 4 4 4 6 4 4 6 a + a - a + 4 a z - 3 a z + a z - a z + 3 a z - 2 a z + a z
In[14]:=	Kauffman[Knot[11, NonAlternating, 159]][a, z]
Out[14]=	2 4 6 5 7 2 2 4 2 6 2 8 2 10 2 -a + a + a - 2 a z - 2 a z - a z - 6 a z - 5 a z + a z + a z + 3 3 3 5 3 7 3 9 3 2 4 4 4 > a z + a z + 6 a z + 13 a z + 7 a z + 5 a z + 12 a z + 6 4 8 4 10 4 3 5 5 5 7 5 9 5 > 13 a z + 4 a z - 2 a z + a z - 9 a z - 21 a z - 11 a z - 4 6 6 6 8 6 10 6 3 7 5 7 7 7 > 8 a z - 21 a z - 12 a z + a z + 2 a z + 2 a z + 4 a z + 9 7 4 8 6 8 8 8 5 9 7 9 > 4 a z + 4 a z + 9 a z + 5 a z + 2 a z + 2 a z
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 159]], Vassiliev[3][Knot[11, NonAlternating, 159]]}
Out[15]=	{2, -3}
In[16]:=	Kh[Knot[11, NonAlternating, 159]][q, t]
Out[16]=	2 4 1 3 1 4 3 6 4 -- + - + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 3 q 19 8 17 7 15 7 15 6 13 6 13 5 11 5 q q t q t q t q t q t q t q t 6 6 6 6 5 6 3 5 > ------ + ----- + ----- + ----- + ----- + ----- + ---- + ---- + q t 11 4 9 4 9 3 7 3 7 2 5 2 5 3 q t q t q t q t q t q t q t q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n159

K11n158 K11n160