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

K11a160 K11a162

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X10,3,11,4 X18,5,19,6 X14,8,15,7 X12,10,13,9 X2,11,3,12 X8,14,9,13 X20,15,21,16 X22,17,1,18 X6,19,7,20 X16,21,17,22

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 10 18 14 12 2 8 20 22 6 16

Alexander Polynomial:

- 2t-3 + 8t-2 - 12t-1 + 13 - 12t + 8t2 - 2t3

Conway Polynomial:

1 + 2z2 - 4z4 - 2z6

Other knots with the same Alexander/Conway Polynomial:

{1014, K11n2, ...}

Determinant and Signature:

{57, -4}

Jones Polynomial:

- q-9 + 2q-8 - 4q-7 + 6q-6 - 7q-5 + 9q-4 - 8q-3 + 7q-2 - 6q-1 + 4 - 2q + q2

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

{...}

A2 (sl(3)) Invariant:

- q-28 - q-26 - q-22 + q-20 + q-18 + q-16 + 3q-14 + q-10 - q-8 - 2q-6 - q-2 + 1 + q2 + q6

HOMFLY-PT Polynomial:

2 + 3z2 + z4 - 3a2 - 5a2z2 - 4a2z4 - a2z6 + a4 - a4z2 - 3a4z4 - a4z6 + 3a6 + 6a6z2 + 2a6z4 - 2a8 - a8z2

Kauffman Polynomial:

2 - 9z2 + 12z4 - 6z6 + z8 + 3az - 12az3 + 19az5 - 11az7 + 2az9 + 3a2 - 20a2z2 + 32a2z4 - 14a2z6 - a2z8 + a2z10 + 5a3z - 16a3z3 + 31a3z5 - 23a3z7 + 5a3z9 + a4 - 13a4z2 + 29a4z4 - 21a4z6 + 2a4z8 + a4z10 + 2a5z - 2a5z3 + 3a5z5 - 8a5z7 + 3a5z9 - 3a6 + 4a6z2 + a6z4 - 9a6z6 + 4a6z8 + 2a7z - a7z3 - 6a7z5 + 4a7z7 - 2a8 + 5a8z2 - 6a8z4 + 4a8z6 + a9z - 2a9z3 + 3a9z5 - a10z2 + 2a10z4 - a11z + a11z3

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

{2, -7}

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=-4 is the signature of 11161. 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 r = 3 r = 4 j = 5                       1 j = 3                     1   j = 1                   3 1   j = -1                 3 1     j = -3               4 3       j = -5             5 4         j = -7           4 3           j = -9         3 5             j = -11       3 4               j = -13     1 3                 j = -15   1 3                   j = -17   1                     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, Alternating, 161]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 161]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 3, 11, 4], X[18, 5, 19, 6], X[14, 8, 15, 7], > X[12, 10, 13, 9], X[2, 11, 3, 12], X[8, 14, 9, 13], X[20, 15, 21, 16], > X[22, 17, 1, 18], X[6, 19, 7, 20], X[16, 21, 17, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 161]]
Out[4]=	GaussCode[1, -6, 2, -1, 3, -10, 4, -7, 5, -2, 6, -5, 7, -4, 8, -11, 9, -3, 10, > -8, 11, -9]
In[5]:=	DTCode[Knot[11, Alternating, 161]]
Out[5]=	DTCode[4, 10, 18, 14, 12, 2, 8, 20, 22, 6, 16]
In[6]:=	alex = Alexander[Knot[11, Alternating, 161]][t]
Out[6]=	2 8 12 2 3 13 - -- + -- - -- - 12 t + 8 t - 2 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 161]][z]
Out[7]=	2 4 6 1 + 2 z - 4 z - 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 14], Knot[11, Alternating, 161], Knot[11, NonAlternating, 2]}
In[9]:=	{KnotDet[Knot[11, Alternating, 161]], KnotSignature[Knot[11, Alternating, 161]]}
Out[9]=	{57, -4}
In[10]:=	J=Jones[Knot[11, Alternating, 161]][q]
Out[10]=	-9 2 4 6 7 9 8 7 6 2 4 - q + -- - -- + -- - -- + -- - -- + -- - - - 2 q + 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, Alternating, 161]}
In[12]:=	A2Invariant[Knot[11, Alternating, 161]][q]
Out[12]=	-28 -26 -22 -20 -18 -16 3 -10 -8 2 -2 1 - q - q - q + q + q + q + --- + q - q - -- - q + 14 6 q q 2 6 > q + q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 161]][a, z]
Out[13]=	2 4 6 8 2 2 2 4 2 6 2 8 2 4 2 - 3 a + a + 3 a - 2 a + 3 z - 5 a z - a z + 6 a z - a z + z - 2 4 4 4 6 4 2 6 4 6 > 4 a z - 3 a z + 2 a z - a z - a z
In[14]:=	Kauffman[Knot[11, Alternating, 161]][a, z]
Out[14]=	2 4 6 8 3 5 7 9 11 2 + 3 a + a - 3 a - 2 a + 3 a z + 5 a z + 2 a z + 2 a z + a z - a z - 2 2 2 4 2 6 2 8 2 10 2 3 > 9 z - 20 a z - 13 a z + 4 a z + 5 a z - a z - 12 a z - 3 3 5 3 7 3 9 3 11 3 4 2 4 > 16 a z - 2 a z - a z - 2 a z + a z + 12 z + 32 a z + 4 4 6 4 8 4 10 4 5 3 5 5 5 > 29 a z + a z - 6 a z + 2 a z + 19 a z + 31 a z + 3 a z - 7 5 9 5 6 2 6 4 6 6 6 8 6 > 6 a z + 3 a z - 6 z - 14 a z - 21 a z - 9 a z + 4 a z - 7 3 7 5 7 7 7 8 2 8 4 8 6 8 > 11 a z - 23 a z - 8 a z + 4 a z + z - a z + 2 a z + 4 a z + 9 3 9 5 9 2 10 4 10 > 2 a z + 5 a z + 3 a z + a z + a z
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 161]], Vassiliev[3][Knot[11, Alternating, 161]]}
Out[15]=	{2, -7}
In[16]:=	Kh[Knot[11, Alternating, 161]][q, t]
Out[16]=	4 4 1 1 1 3 1 3 3 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 5 3 19 7 17 6 15 6 15 5 13 5 13 4 11 4 q q q t q t q t q t q t q t q t 2 4 3 5 4 3 5 3 t 3 t t 2 > ------ + ----- + ----- + ----- + ---- + ---- + --- + --- + -- + 3 q t + 11 3 9 3 9 2 7 2 7 5 3 q q q t q t q t q t q t q t q 3 3 3 5 4 > q t + q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a161

K11a160 K11a162