The Non Alternating Knot 10₁₃₃

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Acknowledgement

KnotPlot

PD Presentation: X₁₄₂₅ X₃₈₄₉ X_14,9,15,10 X_5,13,6,12 X_13,7,14,6 X_18,11,19,12 X_20,15,1,16 X_16,19,17,20 X_10,17,11,18 X₇₂₈₃

Gauss Code: {-1, 10, -2, 1, -4, 5, -10, 2, 3, -9, 6, 4, -5, -3, 7, -8, 9, -6, 8, -7}

DT (Dowker-Thistlethwaite) Code: 4 8 12 2 -14 -18 6 -20 -10 -16

Minimum Braid Representative:

Length is 11, width is 4
Braid index is 4

A Morse Link Presentation:

3D Invariants:

Symmetry Type Unknotting Number 3-Genus Bridge/Super Bridge Index Nakanishi Index

Reversible 1 2 3 / NotAvailable 1

Alexander Polynomial: - t^-2 + 5t^-1 - 7 + 5t - t²

Conway Polynomial: 1 + z² - z⁴

Other knots with the same Alexander/Conway Polynomial: {7₆, ...}

Determinant and Signature: {19, -2}

Jones Polynomial: q^-9 - 2q^-8 + 2q^-7 - 3q^-6 + 3q^-5 - 3q^-4 + 3q^-3 - q^-2 + q^-1

Other knots (up to mirrors) with the same Jones Polynomial: {K11n27, ...}

A2 (sl(3)) Invariant: q^-28 - 2q^-20 - q^-18 - q^-16 + q^-12 + q^-10 + 2q^-8 + q^-6 + q^-2

HOMFLY-PT Polynomial: a² + a²z² + 2a⁴ + 2a⁴z² - 3a⁶ - 3a⁶z² - a⁶z⁴ + a⁸ + a⁸z²

Kauffman Polynomial: - a² + a²z² + a³z³ + 2a⁴ - 3a⁴z² + 2a⁴z⁴ - 4a⁵z + 7a⁵z³ - 4a⁵z⁵ + a⁵z⁷ + 3a⁶ - 6a⁶z² + 6a⁶z⁴ - 4a⁶z⁶ + a⁶z⁸ - 7a⁷z + 16a⁷z³ - 13a⁷z⁵ + 3a⁷z⁷ + a⁸ + a⁸z² - 3a⁸z⁶ + a⁸z⁸ - 3a⁹z + 10a⁹z³ - 9a⁹z⁵ + 2a⁹z⁷ + 3a¹⁰z² - 4a¹⁰z⁴ + a¹⁰z⁶

V₂ and V₃, the type 2 and 3 Vassiliev invariants: {1, 0}

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

t^rq^j r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0

j = -1 1

j = -3 1 1

j = -5 2

j = -7 1 1

j = -9 2 2

j = -11 1 1

j = -13 1 2

j = -15 1 1

j = -17 1

j = -19 1

n Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2 q^-26 - 2q^-25 - q^-24 + 5q^-23 - 3q^-22 - 4q^-21 + 7q^-20 - q^-19 - 6q^-18 + 6q^-17 + q^-16 - 6q^-15 + 3q^-14 + 3q^-13 - 5q^-12 + 4q^-10 - 3q^-9 - q^-8 + 2q^-7 + q^-2

3 q^-51 - 2q^-50 - q^-49 + 2q^-48 + 4q^-47 - 2q^-46 - 7q^-45 + q^-44 + 8q^-43 + 2q^-42 - 8q^-41 - 4q^-40 + 6q^-39 + 5q^-38 - 4q^-37 - 3q^-36 + q^-34 + 5q^-32 - 2q^-31 - 8q^-30 + q^-29 + 14q^-28 - 2q^-27 - 16q^-26 + q^-25 + 20q^-24 - 2q^-23 - 23q^-22 + 2q^-21 + 24q^-20 - 26q^-18 + 21q^-16 + 7q^-15 - 22q^-14 - 5q^-13 + 12q^-12 + 9q^-11 - 9q^-10 - 5q^-9 + 2q^-8 + 6q^-7 - 2q^-6 - q^-5 - q^-4 + 2q^-3

4 q^-84 - 2q^-83 - q^-82 + 2q^-81 + q^-80 + 5q^-79 - 6q^-78 - 5q^-77 + 16q^-74 - 4q^-73 - 7q^-72 - 5q^-71 - 9q^-70 + 19q^-69 - 15q^-65 + 12q^-64 - 7q^-63 - q^-62 + 12q^-61 - 3q^-60 + 14q^-59 - 19q^-58 - 21q^-57 + 12q^-56 + 14q^-55 + 33q^-54 - 18q^-53 - 47q^-52 - 5q^-51 + 21q^-50 + 57q^-49 - 2q^-48 - 64q^-47 - 26q^-46 + 17q^-45 + 73q^-44 + 19q^-43 - 73q^-42 - 43q^-41 + 12q^-40 + 82q^-39 + 34q^-38 - 79q^-37 - 54q^-36 + 10q^-35 + 88q^-34 + 45q^-33 - 83q^-32 - 63q^-31 + 5q^-30 + 87q^-29 + 55q^-28 - 70q^-27 - 65q^-26 - 11q^-25 + 69q^-24 + 58q^-23 - 40q^-22 - 48q^-21 - 22q^-20 + 35q^-19 + 40q^-18 - 12q^-17 - 19q^-16 - 18q^-15 + 10q^-14 + 16q^-13 - 2q^-12 - 3q^-11 - 7q^-10 + 3q^-9 + 4q^-8 - q^-7 - 2q^-5 + q^-4 + q^-3

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]:=
PD[Knot[10, 133]]

Out[2]=
PD[X[1, 4, 2, 5], X[3, 8, 4, 9], X[14, 9, 15, 10], X[5, 13, 6, 12], > X[13, 7, 14, 6], X[18, 11, 19, 12], X[20, 15, 1, 16], X[16, 19, 17, 20], > X[10, 17, 11, 18], X[7, 2, 8, 3]]

In[3]:=
GaussCode[Knot[10, 133]]

Out[3]=
GaussCode[-1, 10, -2, 1, -4, 5, -10, 2, 3, -9, 6, 4, -5, -3, 7, -8, 9, -6, 8, > -7]

In[4]:=
DTCode[Knot[10, 133]]

Out[4]=
DTCode[4, 8, 12, 2, -14, -18, 6, -20, -10, -16]

In[5]:=
br = BR[Knot[10, 133]]

Out[5]=
BR[4, {-1, -1, -1, -2, 1, 1, -2, -3, 2, -3, -3}]

In[6]:=
{First[br], Crossings[br]}

Out[6]=
{4, 11}

In[7]:=
BraidIndex[Knot[10, 133]]

Out[7]=
4

In[8]:=
Show[DrawMorseLink[Knot[10, 133]]]

Out[8]=
-Graphics-

In[9]:=
#[Knot[10, 133]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}

Out[9]=
{Reversible, 1, 2, 3, NotAvailable, 1}

In[10]:=
alex = Alexander[Knot[10, 133]][t]

Out[10]=
-2 5 2 -7 - t + - + 5 t - t t

In[11]:=
Conway[Knot[10, 133]][z]

Out[11]=
2 4 1 + z - z

In[12]:=
Select[AllKnots[], (alex === Alexander[#][t])&]

Out[12]=
{Knot[7, 6], Knot[10, 133]}

In[13]:=
{KnotDet[Knot[10, 133]], KnotSignature[Knot[10, 133]]}

Out[13]=
{19, -2}

In[14]:=
Jones[Knot[10, 133]][q]

Out[14]=
-9 2 2 3 3 3 3 -2 1 q - -- + -- - -- + -- - -- + -- - q + - 8 7 6 5 4 3 q q q q q q q

In[15]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]

Out[15]=
{Knot[10, 133], Knot[11, NonAlternating, 27]}

In[16]:=
A2Invariant[Knot[10, 133]][q]

Out[16]=
-28 2 -18 -16 -12 -10 2 -6 -2 q - --- - q - q + q + q + -- + q + q 20 8 q q

In[17]:=
HOMFLYPT[Knot[10, 133]][a, z]

Out[17]=
2 4 6 8 2 2 4 2 6 2 8 2 6 4 a + 2 a - 3 a + a + a z + 2 a z - 3 a z + a z - a z

In[18]:=
Kauffman[Knot[10, 133]][a, z]

Out[18]=
2 4 6 8 5 7 9 2 2 4 2 6 2 -a + 2 a + 3 a + a - 4 a z - 7 a z - 3 a z + a z - 3 a z - 6 a z + 8 2 10 2 3 3 5 3 7 3 9 3 4 4 > a z + 3 a z + a z + 7 a z + 16 a z + 10 a z + 2 a z + 6 4 10 4 5 5 7 5 9 5 6 6 8 6 > 6 a z - 4 a z - 4 a z - 13 a z - 9 a z - 4 a z - 3 a z + 10 6 5 7 7 7 9 7 6 8 8 8 > a z + a z + 3 a z + 2 a z + a z + a z

In[19]:=
{Vassiliev[2][Knot[10, 133]], Vassiliev[3][Knot[10, 133]]}

Out[19]=
{1, 0}

In[20]:=
Kh[Knot[10, 133]][q, t]

Out[20]=
-3 1 1 1 1 1 1 2 1 q + - + ------ + ------ + ------ + ------ + ------ + ------ + ------ + q 19 8 17 7 15 7 15 6 13 6 13 5 11 5 q t q t q t q t q t q t q t 1 2 2 1 1 2 1 > ------ + ----- + ----- + ----- + ----- + ----- + ---- 11 4 9 4 9 3 7 3 7 2 5 2 3 q t q t q t q t q t q t q t

In[21]:=
ColouredJones[Knot[10, 133], 2][q]

Out[21]=
-26 2 -24 5 3 4 7 -19 6 6 -16 6 q - --- - q + --- - --- - --- + --- - q - --- + --- + q - --- + 25 23 22 21 20 18 17 15 q q q q q q q q 3 3 5 4 3 -8 2 -2 > --- + --- - --- + --- - -- - q + -- + q 14 13 12 10 9 7 q q q q q q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 10₁₃₃

10₁₃₂
10₁₃₄

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q^-26 - 2q^-25 - q^-24 + 5q^-23 - 3q^-22 - 4q^-21 + 7q^-20 - q^-19 - 6q^-18 + 6q^-17 + q^-16 - 6q^-15 + 3q^-14 + 3q^-13 - 5q^-12 + 4q^-10 - 3q^-9 - q^-8 + 2q^-7 + q^-2
3	q^-51 - 2q^-50 - q^-49 + 2q^-48 + 4q^-47 - 2q^-46 - 7q^-45 + q^-44 + 8q^-43 + 2q^-42 - 8q^-41 - 4q^-40 + 6q^-39 + 5q^-38 - 4q^-37 - 3q^-36 + q^-34 + 5q^-32 - 2q^-31 - 8q^-30 + q^-29 + 14q^-28 - 2q^-27 - 16q^-26 + q^-25 + 20q^-24 - 2q^-23 - 23q^-22 + 2q^-21 + 24q^-20 - 26q^-18 + 21q^-16 + 7q^-15 - 22q^-14 - 5q^-13 + 12q^-12 + 9q^-11 - 9q^-10 - 5q^-9 + 2q^-8 + 6q^-7 - 2q^-6 - q^-5 - q^-4 + 2q^-3
4	q^-84 - 2q^-83 - q^-82 + 2q^-81 + q^-80 + 5q^-79 - 6q^-78 - 5q^-77 + 16q^-74 - 4q^-73 - 7q^-72 - 5q^-71 - 9q^-70 + 19q^-69 - 15q^-65 + 12q^-64 - 7q^-63 - q^-62 + 12q^-61 - 3q^-60 + 14q^-59 - 19q^-58 - 21q^-57 + 12q^-56 + 14q^-55 + 33q^-54 - 18q^-53 - 47q^-52 - 5q^-51 + 21q^-50 + 57q^-49 - 2q^-48 - 64q^-47 - 26q^-46 + 17q^-45 + 73q^-44 + 19q^-43 - 73q^-42 - 43q^-41 + 12q^-40 + 82q^-39 + 34q^-38 - 79q^-37 - 54q^-36 + 10q^-35 + 88q^-34 + 45q^-33 - 83q^-32 - 63q^-31 + 5q^-30 + 87q^-29 + 55q^-28 - 70q^-27 - 65q^-26 - 11q^-25 + 69q^-24 + 58q^-23 - 40q^-22 - 48q^-21 - 22q^-20 + 35q^-19 + 40q^-18 - 12q^-17 - 19q^-16 - 18q^-15 + 10q^-14 + 16q^-13 - 2q^-12 - 3q^-11 - 7q^-10 + 3q^-9 + 4q^-8 - q^-7 - 2q^-5 + q^-4 + q^-3

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	PD[Knot[10, 133]]
Out[2]=	PD[X[1, 4, 2, 5], X[3, 8, 4, 9], X[14, 9, 15, 10], X[5, 13, 6, 12], > X[13, 7, 14, 6], X[18, 11, 19, 12], X[20, 15, 1, 16], X[16, 19, 17, 20], > X[10, 17, 11, 18], X[7, 2, 8, 3]]
In[3]:=	GaussCode[Knot[10, 133]]
Out[3]=	GaussCode[-1, 10, -2, 1, -4, 5, -10, 2, 3, -9, 6, 4, -5, -3, 7, -8, 9, -6, 8, > -7]
In[4]:=	DTCode[Knot[10, 133]]
Out[4]=	DTCode[4, 8, 12, 2, -14, -18, 6, -20, -10, -16]
In[5]:=	br = BR[Knot[10, 133]]
Out[5]=	BR[4, {-1, -1, -1, -2, 1, 1, -2, -3, 2, -3, -3}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{4, 11}
In[7]:=	BraidIndex[Knot[10, 133]]
Out[7]=	4
In[8]:=	Show[DrawMorseLink[Knot[10, 133]]]

Out[8]=	-Graphics-
In[9]:=	#[Knot[10, 133]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 1, 2, 3, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 133]][t]
Out[10]=	-2 5 2 -7 - t + - + 5 t - t t
In[11]:=	Conway[Knot[10, 133]][z]
Out[11]=	2 4 1 + z - z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[7, 6], Knot[10, 133]}
In[13]:=	{KnotDet[Knot[10, 133]], KnotSignature[Knot[10, 133]]}
Out[13]=	{19, -2}
In[14]:=	Jones[Knot[10, 133]][q]
Out[14]=	-9 2 2 3 3 3 3 -2 1 q - -- + -- - -- + -- - -- + -- - q + - 8 7 6 5 4 3 q q q q q q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] \|\| (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[10, 133], Knot[11, NonAlternating, 27]}
In[16]:=	A2Invariant[Knot[10, 133]][q]
Out[16]=	-28 2 -18 -16 -12 -10 2 -6 -2 q - --- - q - q + q + q + -- + q + q 20 8 q q
In[17]:=	HOMFLYPT[Knot[10, 133]][a, z]
Out[17]=	2 4 6 8 2 2 4 2 6 2 8 2 6 4 a + 2 a - 3 a + a + a z + 2 a z - 3 a z + a z - a z
In[18]:=	Kauffman[Knot[10, 133]][a, z]
Out[18]=	2 4 6 8 5 7 9 2 2 4 2 6 2 -a + 2 a + 3 a + a - 4 a z - 7 a z - 3 a z + a z - 3 a z - 6 a z + 8 2 10 2 3 3 5 3 7 3 9 3 4 4 > a z + 3 a z + a z + 7 a z + 16 a z + 10 a z + 2 a z + 6 4 10 4 5 5 7 5 9 5 6 6 8 6 > 6 a z - 4 a z - 4 a z - 13 a z - 9 a z - 4 a z - 3 a z + 10 6 5 7 7 7 9 7 6 8 8 8 > a z + a z + 3 a z + 2 a z + a z + a z
In[19]:=	{Vassiliev[2][Knot[10, 133]], Vassiliev[3][Knot[10, 133]]}
Out[19]=	{1, 0}
In[20]:=	Kh[Knot[10, 133]][q, t]
Out[20]=	-3 1 1 1 1 1 1 2 1 q + - + ------ + ------ + ------ + ------ + ------ + ------ + ------ + q 19 8 17 7 15 7 15 6 13 6 13 5 11 5 q t q t q t q t q t q t q t 1 2 2 1 1 2 1 > ------ + ----- + ----- + ----- + ----- + ----- + ---- 11 4 9 4 9 3 7 3 7 2 5 2 3 q t q t q t q t q t q t q t
In[21]:=	ColouredJones[Knot[10, 133], 2][q]
Out[21]=	-26 2 -24 5 3 4 7 -19 6 6 -16 6 q - --- - q + --- - --- - --- + --- - q - --- + --- + q - --- + 25 23 22 21 20 18 17 15 q q q q q q q q 3 3 5 4 3 -8 2 -2 > --- + --- - --- + --- - -- - q + -- + q 14 13 12 10 9 7 q q q q q q

The Non Alternating Knot 10133

The Non Alternating Knot 10₁₃₃