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

K11n169 K11n171

Knotscape

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DrawMorseLink

PD Presentation:

X6271 X10,3,11,4 X16,5,17,6 X12,8,13,7 X18,9,19,10 X2,11,3,12 X13,22,14,1 X15,20,16,21 X4,17,5,18 X8,19,9,20 X21,14,22,15

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

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

Alexander Polynomial:

- 3t-2 + 16t-1 - 25 + 16t - 3t2

Conway Polynomial:

1 + 4z2 - 3z4

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{63, -2}

Jones Polynomial:

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

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

{...}

A2 (sl(3)) Invariant:

- 2q-26 - 2q-24 + 2q-22 - q-20 + q-18 + 3q-16 - q-14 + 2q-12 - q-10 + q-8 + q-6 - 2q-4 + 3q-2 - 1 - q2 + q4

HOMFLY-PT Polynomial:

z2 + a2 + a2z2 - a2z4 - a4 - 2a4z2 - 2a4z4 + 3a6 + 4a6z2 - 2a8

Kauffman Polynomial:

- z2 + z4 - 3az3 + 3az5 - a2 + 3a2z2 - 6a2z4 + 5a2z6 - a3z + 8a3z3 - 10a3z5 + 6a3z7 - a4 + 3a4z2 - 4a4z6 + 4a4z8 - a5z + 10a5z3 - 15a5z5 + 6a5z7 + a5z9 - 3a6 + 2a6z2 + 3a6z4 - 9a6z6 + 5a6z8 + 5a7z - 9a7z3 + a7z5 + a7z9 - 2a8 + 3a8z2 - 4a8z4 + a8z8 + 5a9z - 8a9z3 + 3a9z5

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

{4, -9}

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 11170. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

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

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, 170]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 170]]
Out[3]=	PD[X[6, 2, 7, 1], X[10, 3, 11, 4], X[16, 5, 17, 6], X[12, 8, 13, 7], > X[18, 9, 19, 10], X[2, 11, 3, 12], X[13, 22, 14, 1], X[15, 20, 16, 21], > X[4, 17, 5, 18], X[8, 19, 9, 20], X[21, 14, 22, 15]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 170]]
Out[4]=	GaussCode[1, -6, 2, -9, 3, -1, 4, -10, 5, -2, 6, -4, -7, 11, -8, -3, 9, -5, 10, > 8, -11, 7]
In[5]:=	DTCode[Knot[11, NonAlternating, 170]]
Out[5]=	DTCode[6, 10, 16, 12, 18, 2, -22, -20, 4, 8, -14]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 170]][t]
Out[6]=	3 16 2 -25 - -- + -- + 16 t - 3 t 2 t t
In[7]:=	Conway[Knot[11, NonAlternating, 170]][z]
Out[7]=	2 4 1 + 4 z - 3 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, NonAlternating, 170]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 170]], KnotSignature[Knot[11, NonAlternating, 170]]}
Out[9]=	{63, -2}
In[10]:=	J=Jones[Knot[11, NonAlternating, 170]][q]
Out[10]=	2 4 7 10 10 11 9 6 -3 - -- + -- - -- + -- - -- + -- - -- + - + 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, 170]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 170]][q]
Out[12]=	2 2 2 -20 -18 3 -14 2 -10 -8 -6 2 -1 - --- - --- + --- - q + q + --- - q + --- - q + q + q - -- + 26 24 22 16 12 4 q q q q q q 3 2 4 > -- - q + q 2 q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 170]][a, z]
Out[13]=	2 4 6 8 2 2 2 4 2 6 2 2 4 4 4 a - a + 3 a - 2 a + z + a z - 2 a z + 4 a z - a z - 2 a z
In[14]:=	Kauffman[Knot[11, NonAlternating, 170]][a, z]
Out[14]=	2 4 6 8 3 5 7 9 2 2 2 -a - a - 3 a - 2 a - a z - a z + 5 a z + 5 a z - z + 3 a z + 4 2 6 2 8 2 3 3 3 5 3 7 3 > 3 a z + 2 a z + 3 a z - 3 a z + 8 a z + 10 a z - 9 a z - 9 3 4 2 4 6 4 8 4 5 3 5 5 5 > 8 a z + z - 6 a z + 3 a z - 4 a z + 3 a z - 10 a z - 15 a z + 7 5 9 5 2 6 4 6 6 6 3 7 5 7 > a z + 3 a z + 5 a z - 4 a z - 9 a z + 6 a z + 6 a z + 4 8 6 8 8 8 5 9 7 9 > 4 a z + 5 a z + a z + a z + a z
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 170]], Vassiliev[3][Knot[11, NonAlternating, 170]]}
Out[15]=	{4, -9}
In[16]:=	Kh[Knot[11, NonAlternating, 170]][q, t]
Out[16]=	3 4 2 2 2 5 2 5 5 5 -- + - + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 3 q 17 7 15 6 13 6 13 5 11 5 11 4 9 4 9 3 q q t q t q t q t q t q t q t q t 5 6 5 3 6 t 3 2 > ----- + ----- + ----- + ---- + ---- + - + 2 q t + q t 7 3 7 2 5 2 5 3 q q t q t q t q t q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n170

K11n169 K11n171