8 4

8_3

8_5

Visit 8 4's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 8 4's page at Knotilus! Visit 8 4's page at the original Knot Atlas!

8 4 Quick Notes

Somewhat symmetric representation

Knot presentations

Planar diagram presentation	X6271 X14,10,15,9 X10,3,11,4 X2,13,3,14 X12,5,13,6 X16,8,1,7 X4,11,5,12 X8,16,9,15
Gauss code	1, -4, 3, -7, 5, -1, 6, -8, 2, -3, 7, -5, 4, -2, 8, -6
Dowker-Thistlethwaite code	6 10 12 16 14 4 2 8
Conway Notation	[413]

Three dimensional invariants

Symmetry type Reversible Unknotting number 2 3-genus 2 Bridge index 2 Super bridge index $\{3,5\}$ Nakanishi index 1 Maximal Thurston-Bennequin number [-7][-3] Hyperbolic Volume 5.50049 A-Polynomial See Data:8 4/A-polynomial

[edit Notes for 8 4's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus $1$ Topological 4 genus $1$ Concordance genus $2$ Rasmussen s-Invariant 2

[edit Notes for 8 4's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-2t^{2}+5t-5+5t^{-1}-2t^{-2}$
Conway polynomial	$-2z^{4}-3z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 19, -2 }
Jones polynomial	$q^{3}-q^{2}+2q-3+3q^{-1}-3q^{-2}+3q^{-3}-2q^{-4}+q^{-5}$
HOMFLY-PT polynomial (db, data sources)	$z^{2}a^{4}+a^{4}-z^{4}a^{2}-2z^{2}a^{2}-z^{4}-3z^{2}-2+z^{2}a^{-2}+2a^{-2}$
Kauffman polynomial (db, data sources)	$az^{7}+z^{7}a^{-1}+2a^{2}z^{6}+z^{6}a^{-2}+3z^{6}+3a^{3}z^{5}-az^{5}-4z^{5}a^{-1}+3a^{4}z^{4}-3a^{2}z^{4}-5z^{4}a^{-2}-11z^{4}+2a^{5}z^{3}-5a^{3}z^{3}-3az^{3}+4z^{3}a^{-1}+a^{6}z^{2}-3a^{4}z^{2}-a^{2}z^{2}+7z^{2}a^{-2}+10z^{2}+2a^{3}z+az-za^{-1}+a^{4}-2a^{-2}-2$
The A2 invariant	$q^{16}+q^{10}+q^{6}-q^{4}-q^{2}-1-q^{-2}+q^{-4}+q^{-6}+q^{-8}+q^{-10}$
The G2 invariant	$q^{86}-q^{84}+q^{82}-q^{80}-q^{74}+3q^{72}-2q^{70}+2q^{68}-2q^{66}+q^{62}-2q^{60}+3q^{58}-2q^{56}+q^{54}+q^{50}+2q^{48}-q^{46}+2q^{44}+2q^{38}-q^{36}+q^{34}+2q^{32}-2q^{30}+3q^{28}-3q^{26}+2q^{22}-6q^{20}+5q^{18}-5q^{16}+2q^{12}-4q^{10}+3q^{8}-4q^{6}+q^{4}-q^{2}-2+q^{-2}-2q^{-4}+q^{-6}+2q^{-8}-2q^{-10}+q^{-12}-q^{-14}+q^{-16}+3q^{-18}-4q^{-20}+4q^{-22}-2q^{-24}+q^{-26}+4q^{-28}-4q^{-30}+4q^{-32}-q^{-34}+q^{-36}+q^{-38}-2q^{-40}+2q^{-42}+q^{-46}$

Further Quantum Invariants

Further quantum knot invariants for 8_4.

A1 Invariants.

Weight	Invariant
1	$q^{11}-q^{9}+q^{7}-q^{-1}+q^{-3}+q^{-7}$
2	$q^{30}-q^{28}+2q^{24}-2q^{22}+q^{18}-2q^{16}+q^{12}-q^{8}+q^{6}+2q^{4}+2q^{-2}-2q^{-6}+q^{-8}-2q^{-12}+q^{-14}+q^{-16}-q^{-18}+q^{-22}$
3	$q^{57}-q^{55}+q^{51}-q^{47}-q^{45}+q^{43}-q^{41}-q^{39}+2q^{37}+q^{35}-2q^{33}-q^{31}+3q^{29}+2q^{27}-q^{25}-2q^{23}-q^{21}+2q^{19}+2q^{17}-3q^{13}+3q^{9}-3q^{5}-q^{3}+2q+q^{-1}-q^{-3}-q^{-5}+2q^{-7}+2q^{-9}-2q^{-13}+q^{-15}+3q^{-17}+q^{-19}-3q^{-21}-2q^{-23}+2q^{-25}+2q^{-27}-q^{-29}-3q^{-31}+2q^{-35}+q^{-37}-q^{-39}-q^{-41}+q^{-45}$
4	$q^{92}-q^{90}+q^{86}-q^{84}+q^{82}-2q^{80}-2q^{74}+4q^{72}-2q^{66}-4q^{64}+5q^{62}+4q^{60}+2q^{58}-5q^{56}-8q^{54}+3q^{52}+7q^{50}+6q^{48}-2q^{46}-9q^{44}-3q^{42}+2q^{40}+5q^{38}+3q^{36}-2q^{34}-3q^{32}-5q^{30}+5q^{26}+4q^{24}-q^{22}-6q^{20}-3q^{18}+3q^{16}+6q^{14}+q^{12}-4q^{10}-2q^{8}+4q^{6}+6q^{4}-q^{2}-3-2q^{-2}+3q^{-4}+5q^{-6}-2q^{-8}-4q^{-10}-5q^{-12}+6q^{-16}+q^{-18}-5q^{-22}-4q^{-24}+4q^{-26}+4q^{-28}+5q^{-30}-2q^{-32}-6q^{-34}-2q^{-36}+q^{-38}+7q^{-40}+3q^{-42}-3q^{-44}-4q^{-46}-4q^{-48}+3q^{-50}+4q^{-52}+2q^{-54}-q^{-56}-5q^{-58}-q^{-60}+q^{-62}+2q^{-64}+2q^{-66}-q^{-68}-q^{-70}-q^{-72}+q^{-76}$
5	$q^{135}-q^{133}+q^{129}-q^{127}-q^{121}-q^{119}+q^{115}+2q^{113}+2q^{111}-q^{109}-5q^{107}-4q^{105}+4q^{103}+8q^{101}+5q^{99}-3q^{97}-12q^{95}-8q^{93}+5q^{91}+16q^{89}+11q^{87}-4q^{85}-17q^{83}-17q^{81}-q^{79}+18q^{77}+20q^{75}+4q^{73}-14q^{71}-21q^{69}-11q^{67}+7q^{65}+19q^{63}+14q^{61}-q^{59}-11q^{57}-13q^{55}-7q^{53}+4q^{51}+11q^{49}+12q^{47}+4q^{45}-3q^{43}-11q^{41}-11q^{39}+10q^{35}+11q^{33}+3q^{31}-9q^{29}-11q^{27}-3q^{25}+7q^{23}+9q^{21}+2q^{19}-7q^{17}-7q^{15}+8q^{11}+8q^{9}-2q^{7}-10q^{5}-10q^{3}+q+12q^{-1}+12q^{-3}-11q^{-7}-13q^{-9}-2q^{-11}+11q^{-13}+16q^{-15}+7q^{-17}-5q^{-19}-13q^{-21}-11q^{-23}+2q^{-25}+12q^{-27}+11q^{-29}+4q^{-31}-7q^{-33}-13q^{-35}-10q^{-37}+q^{-39}+9q^{-41}+10q^{-43}+6q^{-45}-4q^{-47}-11q^{-49}-10q^{-51}-q^{-53}+7q^{-55}+11q^{-57}+8q^{-59}-q^{-61}-9q^{-63}-10q^{-65}-4q^{-67}+4q^{-69}+9q^{-71}+8q^{-73}+q^{-75}-5q^{-77}-8q^{-79}-5q^{-81}+q^{-83}+5q^{-85}+6q^{-87}+3q^{-89}-2q^{-91}-5q^{-93}-3q^{-95}-q^{-97}+q^{-99}+3q^{-101}+2q^{-103}-q^{-107}-q^{-109}-q^{-111}+q^{-115}$
6	$q^{186}-q^{184}+q^{180}-q^{178}-q^{174}+q^{172}-2q^{170}-q^{168}+3q^{166}+2q^{162}+q^{160}-q^{158}-7q^{156}-3q^{154}+6q^{152}+5q^{150}+7q^{148}+q^{146}-7q^{144}-16q^{142}-5q^{140}+13q^{138}+15q^{136}+13q^{134}-4q^{132}-21q^{130}-28q^{128}-6q^{126}+24q^{124}+30q^{122}+22q^{120}-6q^{118}-35q^{116}-42q^{114}-15q^{112}+26q^{110}+43q^{108}+38q^{106}+8q^{104}-32q^{102}-52q^{100}-35q^{98}+5q^{96}+37q^{94}+49q^{92}+31q^{90}-5q^{88}-38q^{86}-44q^{84}-23q^{82}+3q^{80}+28q^{78}+35q^{76}+24q^{74}+q^{72}-19q^{70}-25q^{68}-24q^{66}-10q^{64}+9q^{62}+26q^{60}+25q^{58}+12q^{56}-8q^{54}-25q^{52}-25q^{50}-11q^{48}+13q^{46}+22q^{44}+18q^{42}-16q^{38}-17q^{36}-7q^{34}+12q^{32}+16q^{30}+7q^{28}-9q^{26}-16q^{24}-9q^{22}+5q^{20}+22q^{18}+20q^{16}+2q^{14}-19q^{12}-24q^{10}-15q^{8}+6q^{6}+28q^{4}+30q^{2}+10-18q^{-2}-30q^{-4}-28q^{-6}-6q^{-8}+24q^{-10}+38q^{-12}+24q^{-14}-6q^{-16}-27q^{-18}-37q^{-20}-23q^{-22}+7q^{-24}+34q^{-26}+35q^{-28}+13q^{-30}-9q^{-32}-31q^{-34}-33q^{-36}-15q^{-38}+15q^{-40}+32q^{-42}+28q^{-44}+17q^{-46}-9q^{-48}-27q^{-50}-30q^{-52}-12q^{-54}+7q^{-56}+19q^{-58}+28q^{-60}+16q^{-62}-q^{-64}-20q^{-66}-23q^{-68}-18q^{-70}-7q^{-72}+14q^{-74}+21q^{-76}+21q^{-78}+7q^{-80}-5q^{-82}-18q^{-84}-23q^{-86}-11q^{-88}+q^{-90}+15q^{-92}+17q^{-94}+17q^{-96}+4q^{-98}-11q^{-100}-15q^{-102}-15q^{-104}-6q^{-106}+2q^{-108}+14q^{-110}+14q^{-112}+7q^{-114}-7q^{-118}-10q^{-120}-11q^{-122}-q^{-124}+4q^{-126}+7q^{-128}+7q^{-130}+4q^{-132}-6q^{-136}-4q^{-138}-3q^{-140}-q^{-142}+q^{-144}+3q^{-146}+3q^{-148}-q^{-154}-q^{-156}-q^{-158}+q^{-162}$

A2 Invariants.

Weight	Invariant
1,0	$q^{16}+q^{10}+q^{6}-q^{4}-q^{2}-1-q^{-2}+q^{-4}+q^{-6}+q^{-8}+q^{-10}$
1,1	$q^{44}-2q^{42}+2q^{40}-2q^{38}+5q^{36}-4q^{34}+4q^{32}-4q^{30}+5q^{28}-4q^{26}+2q^{24}-4q^{22}+2q^{20}-2q^{18}-2q^{16}+2q^{14}-6q^{12}+10q^{10}-10q^{8}+16q^{6}-10q^{4}+16q^{2}-8+8q^{-2}-7q^{-4}-6q^{-10}+5q^{-12}-8q^{-14}+10q^{-16}-8q^{-18}+6q^{-20}-4q^{-22}+4q^{-24}+q^{-28}$
2,0	$q^{40}+q^{34}+q^{32}-q^{28}-q^{24}-3q^{22}+q^{18}-q^{14}+2q^{6}+3q^{4}+3q^{2}+2+3q^{-2}-2q^{-6}-2q^{-8}-2q^{-10}-2q^{-12}-2q^{-14}+q^{-18}+q^{-20}+q^{-24}+q^{-26}+q^{-28}$
3,0	$q^{72}+q^{66}+q^{64}+q^{62}-2q^{60}-q^{58}-q^{54}-2q^{52}-3q^{50}+q^{48}+3q^{46}-4q^{42}-4q^{40}+3q^{38}+6q^{36}+4q^{34}-q^{32}+3q^{28}+3q^{26}+q^{24}-2q^{22}+q^{18}-q^{14}-2q^{8}-5q^{6}-3q^{4}-q^{2}-1-3q^{-2}-3q^{-4}+2q^{-6}+5q^{-8}+6q^{-10}+2q^{-12}+3q^{-14}+5q^{-16}+6q^{-18}+3q^{-20}-2q^{-22}-4q^{-24}-3q^{-26}-q^{-28}-2q^{-32}-3q^{-34}-2q^{-36}+2q^{-40}+q^{-42}-q^{-46}+q^{-50}+q^{-52}+q^{-54}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{36}-q^{34}-q^{32}+2q^{30}-q^{26}+2q^{24}-q^{20}+q^{18}+q^{16}+q^{12}+q^{10}-2q^{6}-q^{4}+q^{2}-2+q^{-4}-q^{-6}+q^{-10}+q^{-14}+2q^{-16}+q^{-20}$
1,0,0	$q^{21}+q^{17}+q^{13}+q^{9}-q^{5}-q^{3}-2q-q^{-1}-q^{-3}+q^{-5}+q^{-7}+2q^{-9}+q^{-11}+q^{-13}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{46}-q^{42}+q^{38}+q^{36}+q^{32}+2q^{30}-q^{26}-2q^{20}+q^{10}+2q^{8}+2q^{4}+3q^{2}+1-2q^{-2}-q^{-4}-2q^{-6}-4q^{-8}-3q^{-10}-q^{-12}+q^{-14}+q^{-16}+3q^{-18}+3q^{-20}+2q^{-22}+q^{-24}+q^{-26}$
1,0,0,0	$q^{26}+q^{22}+q^{20}+q^{16}+q^{12}-q^{6}-q^{4}-2q^{2}-2-q^{-2}-q^{-4}+q^{-6}+q^{-8}+2q^{-10}+2q^{-12}+q^{-14}+q^{-16}$

B2 Invariants.

Weight	Invariant
0,1	$q^{36}-q^{34}+q^{32}-2q^{30}+2q^{28}-q^{26}+2q^{24}+q^{20}+q^{18}-q^{16}+2q^{14}-3q^{12}+3q^{10}-4q^{8}+2q^{6}-3q^{4}+q^{2}-2+q^{-4}-q^{-6}+2q^{-8}-q^{-10}+2q^{-12}-q^{-14}+2q^{-16}+q^{-20}$
1,0	$q^{58}-q^{54}-q^{52}+2q^{48}+q^{46}-q^{44}-q^{42}+2q^{38}-q^{34}-q^{32}+q^{30}+q^{28}-q^{24}+2q^{20}+q^{18}-q^{16}-q^{14}+q^{12}-q^{8}-q^{6}+q^{4}+q^{2}-2q^{-2}+2q^{-6}+q^{-8}-q^{-10}-2q^{-12}+q^{-16}+q^{-18}-q^{-20}-q^{-22}+q^{-24}+2q^{-26}+q^{-34}$

D4 Invariants.

Weight	Invariant
1,0,0,0	$q^{50}-q^{48}-q^{44}+2q^{42}-q^{40}+q^{38}-q^{36}+2q^{34}+q^{30}+q^{24}+2q^{20}-q^{18}+3q^{16}-2q^{14}+3q^{12}-3q^{10}+q^{8}-3q^{6}-3q^{2}-q^{-2}-q^{-4}-q^{-8}+2q^{-10}+2q^{-14}+3q^{-18}+2q^{-22}+q^{-26}$

G2 Invariants.

Weight

Invariant

1,0

$q^{86}-q^{84}+q^{82}-q^{80}-q^{74}+3q^{72}-2q^{70}+2q^{68}-2q^{66}+q^{62}-2q^{60}+3q^{58}-2q^{56}+q^{54}+q^{50}+2q^{48}-q^{46}+2q^{44}+2q^{38}-q^{36}+q^{34}+2q^{32}-2q^{30}+3q^{28}-3q^{26}+2q^{22}-6q^{20}+5q^{18}-5q^{16}+2q^{12}-4q^{10}+3q^{8}-4q^{6}+q^{4}-q^{2}-2+q^{-2}-2q^{-4}+q^{-6}+2q^{-8}-2q^{-10}+q^{-12}-q^{-14}+q^{-16}+3q^{-18}-4q^{-20}+4q^{-22}-2q^{-24}+q^{-26}+4q^{-28}-4q^{-30}+4q^{-32}-q^{-34}+q^{-36}+q^{-38}-2q^{-40}+2q^{-42}+q^{-46}$

.

Computer Talk

The above data is available with the Mathematica package KnotTheory`, as shown in the (simulated) Mathematica session below. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting. This Mathematica session is also available (albeit only for the knot 5_2) as the notebook PolynomialInvariantsSession.nb.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of August 31, 2006, 11:25:27.5625. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["8 4"];

In[4]:=

Alexander[K][t]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]=

$-2t^{2}+5t-5+5t^{-1}-2t^{-2}$

In[5]:=

Conway[K][z]

Out[5]=

$-2z^{4}-3z^{2}+1$

In[6]:=

Alexander[K, 2][t]

KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.

Out[6]=

$\{1\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 19, -2 }

In[8]:=

Jones[K][q]

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[8]=

$q^{3}-q^{2}+2q-3+3q^{-1}-3q^{-2}+3q^{-3}-2q^{-4}+q^{-5}$

In[9]:=

HOMFLYPT[K][a, z]

KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.

Out[9]=

$z^{2}a^{4}+a^{4}-z^{4}a^{2}-2z^{2}a^{2}-z^{4}-3z^{2}-2+z^{2}a^{-2}+2a^{-2}$

In[10]:=

Kauffman[K][a, z]

KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.

Out[10]=

$az^{7}+z^{7}a^{-1}+2a^{2}z^{6}+z^{6}a^{-2}+3z^{6}+3a^{3}z^{5}-az^{5}-4z^{5}a^{-1}+3a^{4}z^{4}-3a^{2}z^{4}-5z^{4}a^{-2}-11z^{4}+2a^{5}z^{3}-5a^{3}z^{3}-3az^{3}+4z^{3}a^{-1}+a^{6}z^{2}-3a^{4}z^{2}-a^{2}z^{2}+7z^{2}a^{-2}+10z^{2}+2a^{3}z+az-za^{-1}+a^{4}-2a^{-2}-2$

Vassiliev invariants

V2 and V3:

(-3, 1)

V2,1 through V6,9:

V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9 $-12$ $8$ $72$ $114$ $54$ $-96$ $-{\frac {496}{3}}$ $-{\frac {160}{3}}$ $-24$ $-288$ $32$ $-1368$ $-648$ $-{\frac {11871}{10}}$ ${\frac {2182}{5}}$ $-{\frac {20902}{15}}$ ${\frac {1055}{6}}$ $-{\frac {3391}{10}}$

V_2,1 through V_6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V₂ and V₃.

Khovanov Homology

The coefficients of the monomials $t^{r}q^{j}$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$). 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 8 4. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \	-4	-3	-2	-1	0	1	2	3	4	χ
7									1	1
5										0
3							2	1		1
1						1				-1
-1					2	2				0
-3				2	2					0
-5			1	1						0
-7		1	2							1
-9		1								-1
-11	1									1

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

${\textrm {Include}}({\textrm {ColouredJonesM.mhtml}})$

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 17, 2005, 14:44:34)...
In[2]:=	Crossings[Knot[8, 4]]
Out[2]=	8
In[3]:=	PD[Knot[8, 4]]
Out[3]=	PD[X[6, 2, 7, 1], X[14, 10, 15, 9], X[10, 3, 11, 4], X[2, 13, 3, 14], X[12, 5, 13, 6], X[16, 8, 1, 7], X[4, 11, 5, 12], X[8, 16, 9, 15]]
In[4]:=	GaussCode[Knot[8, 4]]
Out[4]=	GaussCode[1, -4, 3, -7, 5, -1, 6, -8, 2, -3, 7, -5, 4, -2, 8, -6]
In[5]:=	BR[Knot[8, 4]]
Out[5]=	BR[4, {-1, -1, -1, 2, -1, 2, 3, -2, 3}]
In[6]:=	alex = Alexander[Knot[8, 4]][t]
Out[6]=	2 5 2 -5 - -- + - + 5 t - 2 t 2 t t
In[7]:=	Conway[Knot[8, 4]][z]
Out[7]=	2 4 1 - 3 z - 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[8, 4]}
In[9]:=	{KnotDet[Knot[8, 4]], KnotSignature[Knot[8, 4]]}
Out[9]=	{19, -2}
In[10]:=	J=Jones[Knot[8, 4]][q]
Out[10]=	-5 2 3 3 3 2 3 -3 + q - -- + -- - -- + - + 2 q - q + q 4 3 2 q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[8, 4]}
In[12]:=	A2Invariant[Knot[8, 4]][q]
Out[12]=	-16 -10 -6 -4 -2 2 4 6 8 10 -1 + q + q + q - q - q - q + q + q + q + q
In[13]:=	Kauffman[Knot[8, 4]][a, z]
Out[13]=	2 2 4 z 3 2 7 z 2 2 4 2 -2 - -- + a - - + a z + 2 a z + 10 z + ---- - a z - 3 a z + 2 a 2 a a 3 4 6 2 4 z 3 3 3 5 3 4 5 z 2 4 a z + ---- - 3 a z - 5 a z + 2 a z - 11 z - ---- - 3 a z + a 2 a 5 6 7 4 4 4 z 5 3 5 6 z 2 6 z 7 3 a z - ---- - a z + 3 a z + 3 z + -- + 2 a z + -- + a z a 2 a a
In[14]:=	{Vassiliev[2][Knot[8, 4]], Vassiliev[3][Knot[8, 4]]}
Out[14]=	{0, 1}
In[15]:=	Kh[Knot[8, 4]][q, t]
Out[15]=	2 2 1 1 1 2 1 1 2 2 t -- + - + ------ + ----- + ----- + ----- + ----- + ---- + ---- + --- + 3 q 11 4 9 3 7 3 7 2 5 2 5 3 q q q t q t q t q t q t q t q t 3 2 3 3 7 4 q t + 2 q t + q t + q t