The Alternating Knot 10₇

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Acknowledgement

KnotPlot

PD Presentation: X₁₄₂₅ X_5,14,6,15 X_3,13,4,12 X_13,3,14,2 X_11,20,12,1 X_19,6,20,7 X_7,18,8,19 X_9,16,10,17 X_15,10,16,11 X_17,8,18,9

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

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

Minimum Braid Representative:

Length is 12, width is 5
Braid index is 5

A Morse Link Presentation:

3D Invariants:

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

Reversible 1 2 2 / NotAvailable 1

Alexander Polynomial: - 3t^-2 + 11t^-1 - 15 + 11t - 3t²

Conway Polynomial: 1 - z² - 3z⁴

Other knots with the same Alexander/Conway Polynomial: {K11a59, K11n3, ...}

Determinant and Signature: {43, -2}

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

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

A2 (sl(3)) Invariant: q^-28 + q^-22 - 2q^-20 - q^-18 - q^-14 + q^-12 + q^-8 + q^-6 - q^-4 + 2q^-2 + q⁴

HOMFLY-PT Polynomial: 1 + z² - a²z² - a²z⁴ + a⁴ - a⁴z⁴ - 2a⁶ - 2a⁶z² - a⁶z⁴ + a⁸ + a⁸z²

Kauffman Polynomial: 1 - 2z² + z⁴ - 3az³ + 2az⁵ - 2a²z² - a²z⁴ + 2a²z⁶ + a³z³ - 2a³z⁵ + 2a³z⁷ + a⁴ - 4a⁴z² + 8a⁴z⁴ - 5a⁴z⁶ + 2a⁴z⁸ - 2a⁵z + 6a⁵z³ - 2a⁵z⁵ - a⁵z⁷ + a⁵z⁹ + 2a⁶ - 10a⁶z² + 20a⁶z⁴ - 15a⁶z⁶ + 4a⁶z⁸ - 5a⁷z + 10a⁷z³ - 6a⁷z⁵ - a⁷z⁷ + a⁷z⁹ + a⁸ - 3a⁸z² + 6a⁸z⁴ - 7a⁸z⁶ + 2a⁸z⁸ - 3a⁹z + 8a⁹z³ - 8a⁹z⁵ + 2a⁹z⁷ + 3a¹⁰z² - 4a¹⁰z⁴ + a¹⁰z⁶

V₂ and V₃, the type 2 and 3 Vassiliev invariants: {-1, 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 10₇. 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 r = 1 r = 2

j = 3 1

j = 1 1

j = -1 3 1

j = -3 3 2

j = -5 4 2

j = -7 3 3

j = -9 3 4

j = -11 2 3

j = -13 1 3

j = -15 1 2

j = -17 1

j = -19 1

n Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2 q^-26 - 2q^-25 + 5q^-23 - 7q^-22 - 2q^-21 + 14q^-20 - 11q^-19 - 9q^-18 + 24q^-17 - 11q^-16 - 19q^-15 + 31q^-14 - 7q^-13 - 28q^-12 + 34q^-11 - 3q^-10 - 31q^-9 + 30q^-8 - 24q^-6 + 20q^-5 + q^-4 - 14q^-3 + 11q^-2 - 6 + 5q - 2q³ + q⁴

3 q^-51 - 2q^-50 + 2q^-48 + 2q^-47 - 6q^-46 - 3q^-45 + 9q^-44 + 8q^-43 - 14q^-42 - 13q^-41 + 16q^-40 + 23q^-39 - 17q^-38 - 33q^-37 + 14q^-36 + 40q^-35 - 5q^-34 - 48q^-33 - 2q^-32 + 47q^-31 + 15q^-30 - 46q^-29 - 23q^-28 + 37q^-27 + 35q^-26 - 32q^-25 - 40q^-24 + 21q^-23 + 48q^-22 - 16q^-21 - 49q^-20 + 8q^-19 + 50q^-18 - 50q^-16 + 37q^-14 + 9q^-13 - 33q^-12 - 4q^-11 + 16q^-10 + 8q^-9 - 13q^-8 + 2q^-7 + q^-6 - q^-5 - 3q^-4 + 9q^-3 - q^-2 - 6q^-1 - 3 + 8q + 2q² - 4q³ - 4q⁴ + 4q⁵ + q⁶ - 2q⁸ + q⁹

4 q^-84 - 2q^-83 + 2q^-81 - q^-80 + 3q^-79 - 8q^-78 + q^-77 + 9q^-76 - 2q^-75 + 8q^-74 - 24q^-73 - 4q^-72 + 24q^-71 + 4q^-70 + 23q^-69 - 51q^-68 - 26q^-67 + 34q^-66 + 18q^-65 + 64q^-64 - 70q^-63 - 61q^-62 + 20q^-61 + 15q^-60 + 120q^-59 - 60q^-58 - 74q^-57 - 6q^-56 - 27q^-55 + 149q^-54 - 39q^-53 - 41q^-52 + 2q^-51 - 87q^-50 + 123q^-49 - 45q^-48 + 20q^-47 + 61q^-46 - 128q^-45 + 64q^-44 - 90q^-43 + 71q^-42 + 145q^-41 - 135q^-40 + 5q^-39 - 152q^-38 + 101q^-37 + 219q^-36 - 126q^-35 - 35q^-34 - 202q^-33 + 112q^-32 + 268q^-31 - 107q^-30 - 59q^-29 - 236q^-28 + 104q^-27 + 288q^-26 - 68q^-25 - 58q^-24 - 254q^-23 + 58q^-22 + 269q^-21 - 7q^-20 - 20q^-19 - 242q^-18 - 11q^-17 + 198q^-16 + 37q^-15 + 45q^-14 - 184q^-13 - 62q^-12 + 101q^-11 + 38q^-10 + 87q^-9 - 102q^-8 - 64q^-7 + 27q^-6 + 10q^-5 + 83q^-4 - 38q^-3 - 37q^-2 - 3q^-1 - 11 + 51q - 8q² - 11q³ - 6q⁴ - 15q⁵ + 22q⁶ - 2q⁹ - 8q¹⁰ + 6q¹¹ + q¹³ - 2q¹⁵ + q¹⁶

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, 7]]

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

In[3]:=
GaussCode[Knot[10, 7]]

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

In[4]:=
DTCode[Knot[10, 7]]

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

In[5]:=
br = BR[Knot[10, 7]]

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

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

Out[6]=
{5, 12}

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

Out[7]=
5

In[8]:=
Show[DrawMorseLink[Knot[10, 7]]]

Out[8]=
-Graphics-

In[9]:=
#[Knot[10, 7]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}

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

In[10]:=
alex = Alexander[Knot[10, 7]][t]

Out[10]=
3 11 2 -15 - -- + -- + 11 t - 3 t 2 t t

In[11]:=
Conway[Knot[10, 7]][z]

Out[11]=
2 4 1 - z - 3 z

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

Out[12]=
{Knot[10, 7], Knot[11, Alternating, 59], Knot[11, NonAlternating, 3]}

In[13]:=
{KnotDet[Knot[10, 7]], KnotSignature[Knot[10, 7]]}

Out[13]=
{43, -2}

In[14]:=
Jones[Knot[10, 7]][q]

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

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

Out[15]=
{Knot[10, 7]}

In[16]:=
A2Invariant[Knot[10, 7]][q]

Out[16]=
-28 -22 2 -18 -14 -12 -8 -6 -4 2 4 q + q - --- - q - q + q + q + q - q + -- + q 20 2 q q

In[17]:=
HOMFLYPT[Knot[10, 7]][a, z]

Out[17]=
4 6 8 2 2 2 6 2 8 2 2 4 4 4 6 4 1 + a - 2 a + a + z - a z - 2 a z + a z - a z - a z - a z

In[18]:=
Kauffman[Knot[10, 7]][a, z]

Out[18]=
4 6 8 5 7 9 2 2 2 4 2 1 + a + 2 a + a - 2 a z - 5 a z - 3 a z - 2 z - 2 a z - 4 a z - 6 2 8 2 10 2 3 3 3 5 3 7 3 > 10 a z - 3 a z + 3 a z - 3 a z + a z + 6 a z + 10 a z + 9 3 4 2 4 4 4 6 4 8 4 10 4 5 > 8 a z + z - a z + 8 a z + 20 a z + 6 a z - 4 a z + 2 a z - 3 5 5 5 7 5 9 5 2 6 4 6 6 6 > 2 a z - 2 a z - 6 a z - 8 a z + 2 a z - 5 a z - 15 a z - 8 6 10 6 3 7 5 7 7 7 9 7 4 8 6 8 > 7 a z + a z + 2 a z - a z - a z + 2 a z + 2 a z + 4 a z + 8 8 5 9 7 9 > 2 a z + a z + a z

In[19]:=
{Vassiliev[2][Knot[10, 7]], Vassiliev[3][Knot[10, 7]]}

Out[19]=
{-1, 3}

In[20]:=
Kh[Knot[10, 7]][q, t]

Out[20]=
2 3 1 1 1 2 1 3 2 -- + - + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 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 3 3 4 3 3 4 2 3 t > ------ + ----- + ----- + ----- + ----- + ----- + ---- + ---- + - + q t + 11 4 9 4 9 3 7 3 7 2 5 2 5 3 q q t q t q t q t q t q t q t q t 3 2 > q t

In[21]:=
ColouredJones[Knot[10, 7], 2][q]

Out[21]=
-26 2 5 7 2 14 11 9 24 11 19 31 -6 + q - --- + --- - --- - --- + --- - --- - --- + --- - --- - --- + --- - 25 23 22 21 20 19 18 17 16 15 14 q q q q q q q q q q q 7 28 34 3 31 30 24 20 -4 14 11 3 4 > --- - --- + --- - --- - -- + -- - -- + -- + q - -- + -- + 5 q - 2 q + q 13 12 11 10 9 8 6 5 3 2 q q q q q q q q q q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 10₇

10₆
10₈

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q^-26 - 2q^-25 + 5q^-23 - 7q^-22 - 2q^-21 + 14q^-20 - 11q^-19 - 9q^-18 + 24q^-17 - 11q^-16 - 19q^-15 + 31q^-14 - 7q^-13 - 28q^-12 + 34q^-11 - 3q^-10 - 31q^-9 + 30q^-8 - 24q^-6 + 20q^-5 + q^-4 - 14q^-3 + 11q^-2 - 6 + 5q - 2q³ + q⁴
3	q^-51 - 2q^-50 + 2q^-48 + 2q^-47 - 6q^-46 - 3q^-45 + 9q^-44 + 8q^-43 - 14q^-42 - 13q^-41 + 16q^-40 + 23q^-39 - 17q^-38 - 33q^-37 + 14q^-36 + 40q^-35 - 5q^-34 - 48q^-33 - 2q^-32 + 47q^-31 + 15q^-30 - 46q^-29 - 23q^-28 + 37q^-27 + 35q^-26 - 32q^-25 - 40q^-24 + 21q^-23 + 48q^-22 - 16q^-21 - 49q^-20 + 8q^-19 + 50q^-18 - 50q^-16 + 37q^-14 + 9q^-13 - 33q^-12 - 4q^-11 + 16q^-10 + 8q^-9 - 13q^-8 + 2q^-7 + q^-6 - q^-5 - 3q^-4 + 9q^-3 - q^-2 - 6q^-1 - 3 + 8q + 2q² - 4q³ - 4q⁴ + 4q⁵ + q⁶ - 2q⁸ + q⁹
4	q^-84 - 2q^-83 + 2q^-81 - q^-80 + 3q^-79 - 8q^-78 + q^-77 + 9q^-76 - 2q^-75 + 8q^-74 - 24q^-73 - 4q^-72 + 24q^-71 + 4q^-70 + 23q^-69 - 51q^-68 - 26q^-67 + 34q^-66 + 18q^-65 + 64q^-64 - 70q^-63 - 61q^-62 + 20q^-61 + 15q^-60 + 120q^-59 - 60q^-58 - 74q^-57 - 6q^-56 - 27q^-55 + 149q^-54 - 39q^-53 - 41q^-52 + 2q^-51 - 87q^-50 + 123q^-49 - 45q^-48 + 20q^-47 + 61q^-46 - 128q^-45 + 64q^-44 - 90q^-43 + 71q^-42 + 145q^-41 - 135q^-40 + 5q^-39 - 152q^-38 + 101q^-37 + 219q^-36 - 126q^-35 - 35q^-34 - 202q^-33 + 112q^-32 + 268q^-31 - 107q^-30 - 59q^-29 - 236q^-28 + 104q^-27 + 288q^-26 - 68q^-25 - 58q^-24 - 254q^-23 + 58q^-22 + 269q^-21 - 7q^-20 - 20q^-19 - 242q^-18 - 11q^-17 + 198q^-16 + 37q^-15 + 45q^-14 - 184q^-13 - 62q^-12 + 101q^-11 + 38q^-10 + 87q^-9 - 102q^-8 - 64q^-7 + 27q^-6 + 10q^-5 + 83q^-4 - 38q^-3 - 37q^-2 - 3q^-1 - 11 + 51q - 8q² - 11q³ - 6q⁴ - 15q⁵ + 22q⁶ - 2q⁹ - 8q¹⁰ + 6q¹¹ + q¹³ - 2q¹⁵ + q¹⁶

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

Out[8]=	-Graphics-
In[9]:=	#[Knot[10, 7]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 1, 2, 2, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 7]][t]
Out[10]=	3 11 2 -15 - -- + -- + 11 t - 3 t 2 t t
In[11]:=	Conway[Knot[10, 7]][z]
Out[11]=	2 4 1 - z - 3 z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[10, 7], Knot[11, Alternating, 59], Knot[11, NonAlternating, 3]}
In[13]:=	{KnotDet[Knot[10, 7]], KnotSignature[Knot[10, 7]]}
Out[13]=	{43, -2}
In[14]:=	Jones[Knot[10, 7]][q]
Out[14]=	-9 2 3 5 6 7 7 5 4 -2 + q - -- + -- - -- + -- - -- + -- - -- + - + q 8 7 6 5 4 3 2 q q q q q q q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] \|\| (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[10, 7]}
In[16]:=	A2Invariant[Knot[10, 7]][q]
Out[16]=	-28 -22 2 -18 -14 -12 -8 -6 -4 2 4 q + q - --- - q - q + q + q + q - q + -- + q 20 2 q q
In[17]:=	HOMFLYPT[Knot[10, 7]][a, z]
Out[17]=	4 6 8 2 2 2 6 2 8 2 2 4 4 4 6 4 1 + a - 2 a + a + z - a z - 2 a z + a z - a z - a z - a z
In[18]:=	Kauffman[Knot[10, 7]][a, z]
Out[18]=	4 6 8 5 7 9 2 2 2 4 2 1 + a + 2 a + a - 2 a z - 5 a z - 3 a z - 2 z - 2 a z - 4 a z - 6 2 8 2 10 2 3 3 3 5 3 7 3 > 10 a z - 3 a z + 3 a z - 3 a z + a z + 6 a z + 10 a z + 9 3 4 2 4 4 4 6 4 8 4 10 4 5 > 8 a z + z - a z + 8 a z + 20 a z + 6 a z - 4 a z + 2 a z - 3 5 5 5 7 5 9 5 2 6 4 6 6 6 > 2 a z - 2 a z - 6 a z - 8 a z + 2 a z - 5 a z - 15 a z - 8 6 10 6 3 7 5 7 7 7 9 7 4 8 6 8 > 7 a z + a z + 2 a z - a z - a z + 2 a z + 2 a z + 4 a z + 8 8 5 9 7 9 > 2 a z + a z + a z
In[19]:=	{Vassiliev[2][Knot[10, 7]], Vassiliev[3][Knot[10, 7]]}
Out[19]=	{-1, 3}
In[20]:=	Kh[Knot[10, 7]][q, t]
Out[20]=	2 3 1 1 1 2 1 3 2 -- + - + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 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 3 3 4 3 3 4 2 3 t > ------ + ----- + ----- + ----- + ----- + ----- + ---- + ---- + - + q t + 11 4 9 4 9 3 7 3 7 2 5 2 5 3 q q t q t q t q t q t q t q t q t 3 2 > q t
In[21]:=	ColouredJones[Knot[10, 7], 2][q]
Out[21]=	-26 2 5 7 2 14 11 9 24 11 19 31 -6 + q - --- + --- - --- - --- + --- - --- - --- + --- - --- - --- + --- - 25 23 22 21 20 19 18 17 16 15 14 q q q q q q q q q q q 7 28 34 3 31 30 24 20 -4 14 11 3 4 > --- - --- + --- - --- - -- + -- - -- + -- + q - -- + -- + 5 q - 2 q + q 13 12 11 10 9 8 6 5 3 2 q q q q q q q q q q

The Alternating Knot 107

The Alternating Knot 10₇